Where oh where do I begin... I'll try to keep it short... Haha...
SAAMI claims that, despite the SLIGHTLY differing cartridge dimensions, the two are interchangeable in modern rifles. In general, it takes a VERY tight chamber to notice the difference in the two, and in that case, those chambers could have issues between different brands of brass, let alone between 308 vs 7.62x51mm brass. Technically, SAAMI spec for 308win is is 62,000psi, but only 60,200psi for the 7.62x51mm. In MODERN rifles, or for rifle models built with differently specified chambers, the 7.62x51mm chambers will still withstand the higher pressures of the .308win round without issues. Modern rifles are milled from the same blanks, so the strength is the same.
The REAL issue between the two comes for the reloader. Where the reloader can get in trouble is by using a maximum .308win charge in 7.62x51mm brass. The 7.62x51mm brass is typically thicker, giving it slightly less capacity. A maximum 308win powder charge may create excessive chamber pressures. I've personally seen 3 rifles fail because of this mistake, one as an instant failure on the first shot, and 2 others as gradual failures where the shooter took several shots, then had their rifles fail (one of them shooting a bolt action had to have his firing pin removed from his cheekbone).
All that said, the reason they sell both versions is because the 7.62x51 military surplus ammo, or especially brass has traditionally been cheaper. And since the brass is thicker, it typically lasts longer for the reloader than the thinner .308win commercial brass. So a high volume shooter may choose to shoot 7.62x51mm brass instead of .308win, and naturally, they want the tightest fitting chamber they can get to avoid the brass over stretching and thinning/failing. Naturally, after a single firing, a 7.62x51mm brass will fireform to fit a .308win chamber, and then the use of .308win reloading dies can circumvent the need for a 7.62x51mm rifle. But the point is, if you're shooting high volumes, the 7.62 is cheaper, and purists will buy a 7.62 rifle and 7.62 reloading dies to match their ammo.
Other guys may already have a battle rifle with a 7.62 NATO chamber, like a CETME, M-1A/M-14, AR-10, M-40A1 sniper rifle, etc etc, so they may already have reloading dies and brass for 7.62x51mm, and may not want to convert.
Personally, all of my rifles are .308win chambers. The specs for brass, dies, and chambers are much more controlled across the industry for the .308win than they are for the 7.62x51mm. I've used 7.62x51mm brass in the past since it was cheaper, always keeping in mind to use 7.62x51mm reloading data instead of the .308win data to keep the pressure under control, even though my dies and my chamber were .308win.
So yeah, essentially they're interchangeable, but if you reload or if you plan on shooting a lot of military surplus ammo (not recommended anyway), the 7.62x51mm may be a cheaper option.