The 308 has a higher max pressure, true. Thats the only real difference. But be careful of units PSI not the same as CUP and it gets confusing easily. The difference is not enough to worry about.
The dimensions for the rounds are the same. If you think they are different, then try and buy some 7.62x51 dies. They dont exist. 308 win is the only cartridge offered.
The only dimensional difference is on the guns, not ammo. The range of headspace allowed in the chambers. 7.62x51 has a longer headpsace range allowed before its a reject. Only really old guns, or open bolt machine guns were made to the longer end of the headspace range. These military gas guns have a max headspace 8 thousands longer than SAMMI max (1.6455" vs 1.638") . However ammo is made to the shorter end of the headspace range (1.631"-1.632"), which is exactly with in spec for 308. In fact, some Nato 7.62 ammo is made with even tighter headspace (1.629-1.630)!
This is much ado about nothing. 308/7.62 are interchangable. If you find crappy surplus from 1957 and it does not run good in your 308, then get some better ammo! 308 commercial is fine for either chamber. If you are really paranoid about such things, get some SAMMI headspace gages (1.630 GO, 1.634 NO GO, 1.638 FIELD REJECT) go crazy.
In general, 308win is a tighter chamber than 7.62. You can shoot either ammo out of either gun, but the 7.62 guns tends to be more forgiving, since its looser. 308 tends to be more accurate since the chamber is tighter.
ETA:
http://www.303british.com/id36.html
And DPMS will void your warrenty for reloads too, so take what they say with a grain of salt.
SAMMI puts out a list of dangerous combinations:
http://www.saami.org/specifications_and_information/publications/download/SAAMI_ITEM_211-Unsafe_Arms_and_Ammunition_Combinations.pdf
Notice that 7.62x51 is not listed as unsafe for 308. And then look at .223 and see whats listed... This makes it clear, the 223/5.56 comparison is incorrect.