Confusingly, the substance called adrenaline by most of the world is known as epinephrine to Americans. Similarly, arenaline's cousin, noradrenaline is correspondingly known as norepinephrine. This proliferation of terminology would make this post too confusing, so I'm going to stick with adrenaline and noradrenaline here. (Apologies to American readers!)
[As an aside: why the dual names? I had no idea, but Professor Wikipedia assures me that an American company called Parke-Davis & Co. somehow managed to market a concentrated form of adrenaline under the name... Adrenalin. Since this was a little too close, name-wise, to the generic name for the substance, it was decided to switch the generic name to epinephrine. The choice was actually quite clever, since adrenaline is derived from the Latin ad (above) + renes (the kidney). Epinephrine is just the (ancient) Greek version of this, with epi + nephros meaning exactly the same thing.]
But back to the topic. For one, they are obviously chemically distinct - adrenaline is produced by the body via a modification of noradrenaline. Just what is changed isn't important in the slightest, so I won't go on about this aspect.
Secondly, they are used in slightly different places. Noradrenaline is used in the postganglionic neurones of the sympathetic nervous system. It is also used as a neurotransmitter, which obviously implies that it is generated within the neurones of the brain too. Adrenaline is largely produced by the adrenal medulla, although this structure does also produce some noradrenaline.
Lastly, while related, their functions differ somewhat. You may know that there are several types of receptors for adrenaline. For instance α1 receptors are found on blood vessels and stimulation leads to their constriction. On the other hand, β1 receptors are located on the heart, and their stimulation leads to an increase in the rate and force of the heart's contraction. Adrenaline is rather nonspecific, stimulating α, β1 and β2 receptors more or less equally. By contrast, noradrenaline exerts predominantly α activity, although it does stimulate the β to a lesser degree. Also, as mentioned, noradrenaline acts as a neurotransmitter in the brain too, playing a role in alertness, arousal and reward pathways.
There's considerably more detail out there on this topic, but, again, I can't see its use unless you happen to be doing a PhD!