For me, the biggest advantage of Lisp dialects is that they make it easy to introduce very powerful and far-reaching abstractions - and idiomatic good code encourages these. For some tasks that's pretty awesome, for example anything that has to do with classic symbolic A.I. or expert systems. You abstract every data type your program uses and write small functions that operate on these types. It's similar to classical OOP, except that you're not forced to use OOP for everything but can instead use a more functional style.

Another big advantage is the syntax, because it almost doesn't exist. The language allows you to focus on the semantics whereas the syntax of expressions becomes neglectable. S-expressions are also among the main reasons why abstractions work so well in Lisp.

Finally, and this only applies to CommonLisp: CommonLisp is very complete in terms of language capabilities. It has static and dynamic types, lexical and dynamic scoping, has OOP and allows for functional programming, and so on. It has less artificial limitations than any of the more recent languages. It doesn't attempt to "hold your hand" by e.g. disallowing the use of OOP with multiple inheritance because "it's bad for you", or any such nonsense. Most LISP dialects also have garbage collection, which saves you from crashes and saves a lot of development time.