Allude is coy, to allude is to refer to something in an indirect manner. But elude's favorite thing to do is hide from the cops; it means "to evade." Because the accent is on the second syllable in both words, it's easy to get them mixed up.

To allude is to talk around something, give hints, and generally not say what you really want to say. You allude to something when you don't want to say it outright:

In fact, McCarthy alluded to Williams as being an afterthought in Green Bay's game plan. (Washington Post)

Your film alludes to various versions of the "sleeping beauty" myth — was there a single starting point? (New York Times)

To elude, on the other hand, is to get away. Elude does love hiding from the law, but it can also refer to an idea you can't grasp or cheap health care:

It was a secluded zone with no mobile telephone reception — perfect for eluding law enforcement snooping. (New York Times)

Getting out of England was going to be his only hope of eluding the British authorities. (Spies: The Secret Showdown Between America and Russia)

How the aspiring artist achieved his accomplished technique eludes us. (New York Times)

Good, cheap health care has long eluded America. (Economist)

Allude to something by saying all but what you actually want to say. Elude evades and gets away, like a name that's on the tip of your tongue.