Quoted:
7.62 has a longer head spacing and lower pressure. The .308 is tighter and higher pressure.
I would not fire .308 in the Enfield unless you feel lucky. It might not hurt anything but it could be a disaster as well.
If I remember correctly 7.62 NATO chamber pressure is something like 50,000 PSI, whereas Modern .308 Winchester chamber pressure is about 85,000 PSI, so yeah firing .308 in a 7.62 Chamber would be bad. .308 is roughly the same as a High Pressure PROOF load for 7.62 NATO you can get away with it ONCE IN A WHILE like for hunting, but DO NOT make a hobby out of it, if you must use .308 Winchester do so only to zero if hunting is your objective, and then only what you need to use in the field, though it would be much wiser to hand load a hunting bullet, over a .762 Spec powder charge, otherwise use .308 for EMERGENCY ONLY, as in your position has been overrun, your out of 7.62 NATO, and it's either use .308 Winchester, or fix bayonet.