What is the relationship between electrical resistance and heat generation?
An incandescent light bulb has some amount of resistance (let's say 30
ohms) and generates a large amount of heat. It's my understanding that
a resistor with the same amount of resistance (30 ohms) generates
considerably less heat.

No, that is incorrect. The same power is dissipated providing that they are both fed from the same voltage source. And both will draw the same current. Ohm's law prevails.

If you short out a battery with a piece of copper wire, there is very
little resistance, but a large amount of heat is generated. What
causes these differences?

The two scenarios (lamp/30 Ω and piece of copper wire) are incomparable. The differences are the loading of a very low ohmage piece of copper wire and the internal series resistance of the battery or voltage supply. There is no comparison to be made without considering all the circuit equivalent components and details.

It seems like it must be. It's my understanding that the more
resistance there is in an electrical circuit, the slower a battery
will drain, suggesting that the resistance is somehow decreasing the
speed that energy exits the battery, not just expending all the energy
as heat.

More resistance means less current which equals slower battery drain. However, all the energy that is drained is converted to heat (and some light as per black body radiation).