The most important difference between .308 Winchester and 7.62x51mm NATO is the chamber in the gun. 7.62 NATO chambers are much longer than .308 Winchester, to the point that a NO-GO gauge in .308 Win is still shorter than a 7.62 NATO GO gauge. Firing .308 Winchester in a 7.62 NATO chamber is the equivalent of shooting it in a .308 Win rifle that will close the bolt on a field gauge with plenty of wiggle room left over. This is pretty much the definition of an unsafe gun/ammo combination.
7.62x51mm NATO gets away with these incredibly loose chambers because it has very thick brass. When the case flows under pressure, there are plenty of reserves at the base to ensure that enough brass remains to prevent gasses from rupturing the metal. .308 Winchester is much thinner because it's made in anticipation of a much tighter chamber; it won't have to flow much, so it trades the extra case material for higher capacity. Because of the higher capacity and the civilian powders used in many .308 cases, .308 tends to get better performance than 7.62x51mm NATO.
The performance difference, plus the fact that several sources list 7.62x51mm NATO as rated around 50K PSI and .308 Winchester at 62K PSI max, has led many to conclude that .308 Winchester is loaded to a higher pressure than 7.62 NATO. In reality, the 50K PSI figure came about because of different measurement techniques and mislabeling; it's really a CUP measurement, and 50K CUP translates to about 62K PSI. The difference in performance between .308 Win and 7.62x51mm NATO come from the former's increased case capacity. The real reasons not to use .308 Winchester in a 7.62mm NATO rifle are twofold: First, .308 Winchester can damage a military semi-auto built around the more sedate ballistics and pressure curves of 7.62mm NATO. More importantly, though, 7.62x51mm NATO chambers have far too much headspace to be safe shooting .308 Winchester ammo.