I'm not sure I'm understanding the differences between these two carbines?

603 upper (some 723s seem to be C7's, but the Colt ad shows no bump?)
A1 lower
.625 lightweight barrel
6 hole handguards
Birdcage

So 653=metal stock 723=fiberlite?
What else am I missing?

653 had a 1 / 12" twist rifling, the 723 1 / 7".

The 723 SHOULD have the "C7" upper, but as you pointed out, might not.

I can't find the Colt ad at the moment, but if memory serves me right the 723 would have a delta ring while the 653 would have a slip ring. Of course, posting that here is the best way I know of to find out if that's right.

ETA - And the 653 would have a small rectangle port door (per RBR and others); I don't know about the 723.

Take your piick

653 has an A1 pistol grip and the 723 has an A2 pistol grip.
All minor details, but that's what we're all about here, minor details.
When I showed off my entire collection earlier this year to my 74 year old father and step mother, I realized how minor the details were.
I was surprised that they seemed interested in the minor details.
"What's different about that one?  Oh yeah, now I see it."
Of course they're both collectors too so that helps.
Her: dolls
Him: Knives and handguns

The 723 was basically an "A2" version of the 653, as described in the above answers w/many of the detail changes of the A2 incorporated but the 723 was made in a bewildering variety of parts combos if you go by the Colt operator & repair manuals
653: A1 lower, 603 upper, 14.5" 1/12 barrel w/A1 type lock washer & A1 "birdcage" FH, A1 grip, XM177E2/629 type coated aluminum stock, 2 posiiton tube, delta ring, 6 hole HG's, safe/semi/auto FCG.
723: A1 lower mkd M16A2, also seen w/A2 profile lowers occasionally, uppers are mostly A1 w/deflector (or C7 type) but also occasionally w/A1 uppers & a few A2 uppers!, barrel, 14.5" 1/7 w/peel washer & A2 FH, A2 grip, fiberlite stock w/2 pos tube, delta ring, 6 hole HG's, 723's were offered in both burst & auto FCG configurations & some (most probably late 723's) can even be found w/727/M4 contour barrels!
The 653 seems to be pretty "stable" in features/configuration, the 723 & the following 727 are a bit harder to "pin down" as to features/configuration, going by various Colt catalogs, manuals etc it's hard at times to differentiate between the 723 & 727 as it appears both could be found w/A1 uppers w/deflectors (or in the case of the 723 w/a 'standard" A1 type) or A2 types, both can be found w/thin profile 1/7 barrels or "M4" contour barrels, & both can be found w/either A1 profile lowers or A2 & some of the lowers can even be found w/combinations of both A1 & A2 features!, both 723 & 727 appear though to have been marked as M16A2 regardless of the "profile" of the lowers. These carbines were developed just after the A2 was adopted & it appears that the lowers (A2's) went thru a period where the reinforcements at the rear & front weren't "finalized", most likely the carbines used whatever lowers were available, hence A1 on a lot as the "finalized" type of A2 lowers were in demand for the M16A2 contracts for the Marines & Army which were being being filled at the time the carbines (723 & 727) were being "developed" & marketed, there doesn't seem to have been any major military purchases of the carbines at this time (84-90) just small purchases for specific units/purposes & trials.
The above is just my "take" on it from my search for info/details on the 723 & 727 when I was building my replicas of them. So the end is that the 653 was pretty definitive in it's features as described above compared to the 723, the differences between the 723 & 727 are not so easy (if even possible) to find & list.

ok so:

653:
1/12 twist barrel
Slip Ring
A1 Grip
603 Upper
Aluminum Stock
Small Rectangular Port Door

723:
1/7 twist barrel
Delta Ring
A2 Grip
C7 Upper (more commonly anyway)
Fiberlite Stock
Modern Port Door (?)

Thanks guys.

Both the 653 & 723 used the delta ring, that started in the late 60's w/the XM177E2 & continued on thru the 629, 653 etc, that said, is it possible a 653 may be seen/found w/a straight slip ring, yes, but the delta ring was pretty much standardized on the carbines in the late 60's.
The port doors on most all "723's" I've seen are the A2 style but finding an A1 type on an early 723 is also possible as Colt appears to have used what was "in the bin" for most of these carbines, I have notes on 2 723's which have A1 type doors, 1 is a thin profile barrel the other is an M4 contour/727 barrel!, if doing a very early 723 you could probably get away w/an A1 door, anything later, A2, as mentioned above a lot of the smaller parts appear to have had an overlap in useage between "models". I used A2 doors on both my 723 & 727's. Another small detail is the selector & selector markings, most 723's appear to have had the markings on both sides of the lower like the A2 w/the "tic" marked selector although some may be found w/o the markings on the right &/or a no tic selector. The 653 would only be marked on the left w/a no tic selector.

While I don't claim to know the details, all of the 653 images I've found and the Retro Black Rifle reference page seem to show slip rings.

slobbers over Colt 703

You may be correct there, looking at the carbine guide & the pic shown there the 653 upper does have a slip ring, although that doesn't seem to make "sense" since the XM177E2 went to the delta ring & it was used on all the carbines since, (with the possible exception apparently of the 653 LOL!), it would seem strange that they would go back to the slip ring as the 653 was basically a developement from the XM177E2 w/a slightly longer barrel & no compensator to "solve" the timing & rate of fire "problems" w/the 11 1/2" barrel weapons it would seem strange that they would go back to the older (less "efficient") style ring, even the 629 used the delta ring, but as w/all things Colt it seems there are many anomolies trying to figure all this stuff out LOL! I never built a 653 replica so haven't really "researched" them to any degree, it's difficult to really pin down most of these features on the non-military contract guns, 653, 723, 727 etc as documentation is more difficult to obtain & at times is even different between Colt publications. The guide on the carbines does show a 653 upper w/a slip ring, the pic of the slip ring itself appears to be the same upper, so my question would be, how many 653 uppers have been observed w/slip rings?, it seems that since earlier carbines used the delta & the later carbines used the delta why would the 653 use the old style?, that being said though, the only illustration we have of a 653 shows a slip ring so I assume they did use them.

A Colt carbine with a C7 style upper and M4 barrel is a 725.  It has its own model number.

Will, I could swear I've seen flat slip rings on newer weapons like the 231 or 633?  Maybe I'm crazy, or those were just on RBRD infected repros?

The 725 is the Colt Model number for the C8.  The C8 did not have an M4 profile barrel.  Neither of the subvariants I'm aware of, the 725A and 725B, had that profile either.

I have a factory Colt conversion box that says 725 on the end of it.

A upper with a C7 receiver and M4 barrel came out of it.  It was factory assembled and I removed it NIW so I know it hadn't been tampered with.  We should look into this further.

Can you post a picture of the box?  A conversion kit should have a 2 letter suffix to the model as well.

Oh boy this is going to turn into something interesting LOL! One of the things I've found in trying to "research" different M16/Ar15 variants is that military purchased/accepted/delineated models like the M16A1, M16A2, XM177E2, M4 etc are a bit easier to track, the military generally has a "standard" that's used, on "commercial" models like the 653, 723 727 etc there seems to be a lot more "leeway" as to how they're built & w/what parts, generally speaking if the military wants a barrel assembly for an M4 the barrel weight, contour, FSB etc meets the military "standards" & are pretty well delineated, if there's a change say in weight, contour etc it has to be approved by the military & accepted, ie: the "standard" M4 barrel vs the "heavy" barrel for the M4A1, on the commercial models there's not any regulation other than what Colt assembled, so a very late 723 may have been fitted w/an M4/727 contour barrel on a particular day as that's what was there, I have a number of Colt catalogs & Colt Operators/maintenance/parts catalogs from the late 80's thru the 90's & even trying to use them to figure out what parts are which is difficult. I'm in the process of trying to chart the individual parts used in US military used M16 models from the 601 thru the M4/A4, interesting things are turning up!, the early Colt triangle CH has a different part# (in the beginning the military simply used Colt part #'s!) than the "rounded" charging handle that became standard, yet the early receiver extension tube that had no flats & had to be pinned to the lower used the same part/NSN that the current rifle tubes do, the barrel # for a 1/14 barrel is the same in 1960 as the # in 1966 for the XM16E1 1/12, sometimes when a part was changed the # (either Colt's &/or the NSN) was changed, other times they weren't!, case in point, the forward assist button, the triangle button's part# was the same as the lg round, the lg round w/flat yet the A2 small button shows as a different part#! I haven't been able to figure what would cause the military (or Colt) to issue a new number for a particular part but carry other replacement parts that are obviously different on the same numbers, I'm definitely not "the sharpest tool in the shed" LOL! but there seems to be no rhyme or reason as to how or why this was (or wasn't) done & this is only on the US military models which are built/maintained much closer to a "standard" than most of the commercial models, there seems to be a very large discrepency as to what constitutes a 723, 733, 727, 737 etc in Colt's eyes than we as collectors would like, maybe they're just trying to make it difficult on us!!!!

The largest discrepancies are by far in the commercial/export portion of the 700 series.  During that period from the late 1970s to the early 1990s, Colt seemed to have both entertained a lot of custom configurations, and assembled a lot of weapons at the factory from whatever was close at hand.  

If you have old Colt catalogs and other paper materials then I would vote for trying to find a way to scan them.

I will this weekend when I am with it.

The 653 seems to crop up a lot in photos of foreign militaries and special units.  It seems to have seen a lot of export sales around the same period that the M16A1 was also beginning to be exported in large quantities.  I speculate that the use of the flat slip ring may have been to maintain parts commonality for end users, both foreign and the US military, who also had the M16A1.

ETA:  This of course only raises the question of why they didn't start putting delta rings on A1 rifles...

Since reading all of this and having a hodge podge of parts without a home, I'm considering a 723 build of sorts.  I have an A2 lower in XM gray I got from NDS, have a C7 upper that I can have Victor do in XM Gray and I just ordered the M&A Fiberlite stock kit.  This gets me close but I need a barrel.  Anyone selling 16" pencil barrels or better yet, a 14.5?  I can turn down a 16" locally to 14.7 so I can pin on a regular A2 flash hider.  Am I on the right track here?

ADCO has Daniel Defence 14.5 LW w/ front sight in stock.

Link

Thanks for that.  Should I be looking for a light weight or gov't profile?

Same, same- govt profile is lightweight under the handguards.  It's only on A2 barrels or M4 barrels that they are heavier at the FSB and forward of the FSB.  The DD 14.5" lightweight barrel is a great barrel to use for a 723 style build, but you'll just need to do a perm install of a muzzle device to get the non-NFA length.

Look at the J&T LW CL commando upper .

Look at the J&T LW CL commando barreled  upper . (And see if they would build one around a C7 upper for you )