I can try.
Carbines traditionally were shorter lighter versions of rifles, they gained popularity during WWII, where say for the american troops, for a M1 Garand in 30.06 full size battle rifle you had a M1 carbine in 0.30.
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30.06 vs .30 carbine
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Post WWII
This equation got a little muddled up with intermediate calibers which essenstially were shortened version of full size rifle cartridges, starting with 7.92x33 kurz, 7.62x39 (akm), and then a new cartridge derrived from the .223 the 5.56x45 Nato.
These could function very well as carbines, and were dubbed Assault rifles. They were all the craze back in the 60's but thier shortcommings started to show up post-vietnam. You cant beat physics, when you are working with smaller bullets or smaller charge, you will either get a terminal ballistic penalty or a range penalty. So now most of the armies depending on the type of war they were fighting, had to keep a full size battle rifle handy 7.62x51Nato, G3, Fal's, M14's, etc. (In India's case now the SLR's and now the Sig 7.62x51 rifle)
Coming back to carbines in it's modern iteration, the carbine is supposed to provide intermediate cartridge engeagement ranges, i.e upto 400m, in a light package, in total length of about 30-34". So a lot of rifles can be classifed to discharge this role.
Every AKM with 12" barrel, every AR platform with 12-14" barrel, and all the bullpups with even 18" barrels under the sun can be classified as carbines. But the bullpups with full length barrels might exceed the weight requirements.
Now coming back to the role of carbines and why IA needs them. Shorter Carbines are easier to travel with whether it's in a APC, or a IFV, or while jumping out of an aircraft. It's also easier to run patrols with, it is absolutely essential for urban combat. Screening up a staircase with a 12" barrel M4 or even a SMG is much more effective than running a 16-18" fullsize rifle.
Now coming to your specific query of 7.62x51 rifle vs the carbine.
lets take an example of M4 carbine vs the Sig 7.62x51 rifle. First the Sig is heavier, and runs a larger caliber, in an CT type urban fire fight where you are trying to screen a street or a part of forest cover it's much easier to manuever, and then if you do have to engage, a 5.56x45 Nato has almost negligible recoil impulse, compared to the 7.62x51. If you shoot a 7.62x51nato in close quarters you will most likely loose your sight picture and will have to re adjust to engage again. Now if you are using the same 7.62x51 from a significant distance that no longer is the issue. Full scale battle rifles are designed to engage hostile units from a distance of 200-300m, but more you reduce the stand off distance, you need systems that are more agile.
So it essentially comes down to type of environment you are in and the engament ranges you anticipate.
700-1200m ranges AMR, HMG, GPMG, Specialized Sniper Rifles (338, 416 stuff)
400-700 m ranges - LMG's, General Sniper Rifles, Optics mounted DMR's Bolt action Rifles Full size cartridges 300 Winmag, .308, .243 etc.
300- 600 m Ranges - Battle rifles, Assault rifles (Steyr Aug. M16, etc), Specialized DMR
100-300 m ranges - Assault rifles (AKM, insas, Galils) Carbines,
15-100 ranges - SMG's, PDW, pistol caliber carbines,
0-20 m ranges - Pistols, revolvers.
hope that helps.
One aspect we do have to account for is the continued development in Bullet dynamics and the advent of new cartridges which now bridge the gap between the intermediate and full size battle rifle cartridges like 6.5 Grendel, 6.8 spc and 6.5 Lapua. It is absolutely critical for the large armies around the world to be cognizant of such developments and plan accordingly. I have long argued that 6.5 Grendel is the right step in the future for the Indian forces. Imagine a Carbine that discharges the role of a battle rifle... That's what a modern cartridge can do for the indian forces. With one cartridge platform, remove 5.56 nato, 7.62 nato, 7.62x39 and 7.62x54R from indian services.