The biggest problem between shooting 7.62 NATO in a .308 Winchester headspace barrel or vice versa appears when you reload the brass.
Because the tolerence dimensions are different enough between the two chambers, you will have cartridge cases that will overstretch,(.308 in 7.62 NATO), or be overcompressed,(7.62 in a .308 chamber), which will cause the brass to work harden.
Neither is a major, super duper, big deal unless you reload the cartridge cases.
Because of excessive wear, the cartridge cases can and will begin to crack or even seperate the case head from the case body in as little as three loadings.
This can be alleviated somewhat by annealling the cases after each shooting and before running them through the reloading sizer die.
There is another way around it if you keep all the brass and only load it for that one particular semi automatic rifle.
Gather the brass, make sure all the brass is of the exact same brand and lot, anneal the brass and only neck size it.
This works especially well if you insist on firing commercial specification cartridge cases in a rifle with headspace of 1.634"-1.638" which is a NATO specification.
This doesn't work well at all if you are shooting NATO tolerence brass in a commercial chamber, in this case you will be better served by annealling the brass and full length sizing.