Considering a focused laser beam of 1000mW power, it will burn paper or wood slightly. Focussing the light of an 1000mW LED does not do so. Why is that? Is it because the laser is monochromatic? What makes the difference?
You need a better lens. Seriously, which is to say that the LED is dumping its output power into a large solid angle, so re-collecting it all is rather difficult.
Also, are you certain you're quoting the mean output power of both devices? If that's their peak power, the LED's mean power could be far less if (as with many LEDs) it's pulsed to reduce the internal heat load.
What matters is the power per unit area. The reason the laser can burn is that the laser, being well collimated, can be focused to a much smaller point than the LED. As the power from the LED is spread over a larger area in the source emitter, the optical invariant prevents one from focusing this to a small point while capturing the whole solid angle of emitted power. The original answer is wrong in this regard, there is no lens capable of doing this in such a way as to match the power density of the focused laser.