There are a couple of different reasons to warm up before exercise:
- Increased performance - When you are sitting, you do not have the same cardiac output as you would during exercise, your muscles are at a lower temperature, and capillaries are not necessarily ready to move blood through the muscle. Once you get warmed up (8-15 minutes), capillaries are open, cardiac output to the muscles is increased up to 70% more than at rest, and in addition, the higher temperature is more conducive to oxygen release and better muscle contraction.
- Injury prevention - This is a bit more tenuous, but old animal studies showed warmer muscles took more force to tear than at rest muscles, and some more recent studies show ECG (heart tracing) abnormalities when going from 0-60.
As far as warming up differently, the basic goal should be the same, get the heart rate up, core temperature up, etc. While the method doesn't really matter, the one thing that should be taken into consideration is dynamic stretching of the body parts that will be most involved in the exercise. Mimic the motions of whatever you are going to do in increasing amplitude.
While there is somewhat of a lack of studies on warmup impact on performance, this review of study literature suggests a 79% improvement in performance markers when using a warmup compared to no warmup.