Even though different cartridges may take the same bullets, not all the cartridges will extract the same bullet performance. You simply can’t expect a bullet to act the same way from a .308 Win. as it does from a .30-378 Wthby. Mag. It may be traveling nearly 1,000 fps faster from the big gun, and that changes things radically.
A good example of this is the old Nosler Ballistic Tip bullets. For a long time, many hunters thought they were too "soft" and didn’t like them for big game hunting because they tended to come apart and not exit the animal. However, when used at the lower velocities of single-shot handguns, they performed much differently, acting like a good expansion-controlled big game bullet and penetrating very well. (Nosler has since changed the Ballistic Tip design to "toughen" it up a little, and now many of the rifle shooters who didn’t like it before are embracing the Ballistic Tip as a great, but still relatively soft big game bullet.)
Generally speaking and will all else being equal, the larger the capacity of the case, the heavier the bullet you should use. For example, when hunting deer, a 140-grain bullet is a good choice for the 7mm-08 Rem., but a 160-grain may be far better for the 7mm Rem. Mag. Usually, the heavier the bullet within a given diameter, the tougher it is because the manufacturers expect that the heavier bullets will be used for bigger game and from bigger cases. At magnum impact velocities, the 160-grain may perform similarly to the 140-grain at the lower 7mm-08 impact velocity.
Smaller cases simply don’t have the powder capacity to utilize heavy bullets well. For example, the 200-grain is an excellent weight for the .300 magnums, but trying to use it in a .308 will result in disappointment. You simply cannot push it fast enough for good performance.
On the other hand, the 150-grain is a great bullet in the .308 Win. for deer-sized game, but many 150-grain bullets will come apart on impact from a .300 Wthby. Mag., thus wasting meat and failing to penetrate.
It is always best to match the cartridge to both the game being hunted and the expected shooting distances, and then match the bullet to all three. This, of course, applies to bullet design as well as weight.