Well, you say you've seen both kinds of bows so I don't guess you need a description. Recurves are ancient designs updated with modern materials. Compounds are thoroughly space age designs that have only been with us for the past 40 years.

Basic differences are:

A compound stores more energy per unit of draw weight and once the cam(s) roll over at the end of the draw cycle, the draw weight drops dramatically, anywhere from 65-99% depending on design. The increased energy storage makes for faster arrows. They are, for all practical purposes, universally shot with mechanical release aids, sights and mechanical arrow rests. There is a wide variety of accessories that you can add to the bow to make them more pleasant to shoot. It's not unusual for the bow's collection of accessories to equal or surpass the price of the bow.

A recurve stores less energy per unit of draw weight than the compound. It therefore shoots much slower arrows than the compound. Recurves are almost always shot with fingers release. Though you can certainly use sights on a recurve if you want, most do not. Instead, traditional shooters use a variety of barebow (without sights) aiming methods ranging from 'instinctive' - just look, point and shoot - to various methods of using the point of the arrow as an aiming device. About the only accessories a traditional shooter needs are a shooting glove or tab, an armguard, quiver and a bow stringer, maybe string silencers and a bow case.

Just as a compound forces you to be a better hunter than you need be with a rifle, a recurve forces you to be a better hunter than you need be with a compound. The recurve also demands more time, energy and effort to gain and maintain bowhunting proficiency than a compound does. But recurves are so much fun to shoot that it really doesn't seem like work. So it doesn't seem to be nearly as difficult to find the ability to free up shooting time as it appears to be with the compound.

And yeah, I'm biased. I have used compounds for years for tournament shooting, but I love the recurve and longbow for bowhunting.