An LED just happens to be very poor at dissipating heat beyond it's design limits, but it still obeys Ohm's Law.
Absolutely, positively, NOT true. There is a (positive) relationship between applied voltage and forward current, yes, however it is not linear like Ohm's law would imply. LEDs are not resistors; they are (surprise) diodes. Ohm's law applies to resistors. A standard light bulb is approximately a resistor. If you want to model and LED, you need a model for a diode. A suitable model is the Shockley diode equation. Long story short, the relationship is exponential. That is, current increases exponentially with voltage. This is why LEDs are relatively sensitive to minor fluctuations or differences in voltage and are always driven in some current limited (either constant current, with a resistor, or from an inherently limited supply such as a battery chain) fashion.
The purpose of the resistor when lighting an LED like is commonly done is to attempt to swamp the LED's exponential I/V relationship with the linear nature of the resistor. That is, one wants to allow to resistor's behavior to "make up" for minor imperfections in power supply. Most LEDs also have inconvenient "typical" forward characteristics in terms of voltage (1.8-2.5V is common for non-superbright types), so the resistor also lets you drive it from a more "normal" rail like 3.3V or 5V. The idea is that one treats the diode as a constant voltage device (above a certain point, they are very close) and then sets the current using the resistor. A constant current power supply (an item commonly found on EE lab benches) can also be used, but a resistor and constant voltage supply is a lot simpler and easier to come by. Note RandyT's comment about even relatively small differences in power supply specifications substantially affecting lifetime: that's the exponential behavior of the LED at work.
In this case, your LED is actually an incandescent replacement module and therefore has everything it needs built in. Just apply 5V at the shown polarity. Your 12V lamp is an incandescent light bulb. Apply 12V AC or DC as you please.
I really feel like I've said this sort of things several times over the past week. Perhaps a sticky FAQ on LEDs is in order?