'dis-' means badly or wrongly; 'un-' means 'not'. You would perhaps be dissatisfied with something that exists but was bad (e.g. with a meal that tasted horrible) but unsatisfied with a LACK of something (a meal that was nice but too small). In many cases that difference isn't important and you can use either. However, we don't say 'dissatisfactory' often - 'unsatisfactory' is much more common.

These are VERY similar in meaning and can, I think, be used interchangeably in speech and writing. Dissatisfied is perhaps a little stronger in tone (a dissatisfied person is perhaps a little more angry/upset/saddened than an unsatisfied person...who is simply not satisfied) but the distinction is very subtle.