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It can be inspiring to work with rich Lisp development environments and see what people build when it's so easy to work with source code. It's one of the reasons that so many experiments in language design have emerged from Lisp.
In 2021, this is an argument both for and against Lisp: Lisp implementations are sufficiently fast, so Lisp is best. Modern languages are powerful, so they are best.
You have two main dialects to choose between: Scheme and Common Lisp. They each have advantages and disadvantages, but the differences between them are tiny compared to the differences between them and other languages, so if you want to start learning Lisp, it doesn't matter which you choose.
A Lisp program tends to provide a much clearer mapping between your ideas about how the program works and the code you actually write.
Lisp is a large and complex language with a large and complex runtime to support it. For that reason, Lisp is best suited to large and complicated problems.
Now, a complex problem isn't the same as a complicated one. A complex problem is one with a lot of small details, but which isn't hard. Writing an airline booking system is a complex business, but with enough money and programmers it isn't hard. Get the difference?
A complicated problem is one which is convoluted, one where traditional divide and conquer doesn't work. Controlling a robot, or working with data that isn't tabular (languages, for example), or highly dynamic situations.
Lisp is really well suited to problems where the solution must be expandable; the classic example is the emacs text editor. It is fully programmable, and thus a programming environment in it's own right.
In his famous book PAIP, Norvig says that Lisp is ideal for exploratory programming. That is, programming a solution to a problem that isn't fully understood (as opposed to an on-line booking system). In other words: Complicated problems.
Furthermore, learning Lisp will remind you of something fundamental that has been forgotten: The difference between Von Neumann and Turing. As we know, Turing's model of computation is an interesting theoretical model, but useless as a model for designing computers. Von Neumann, on the other hand, designed a model of how computers and computation were to execute: The Von Neumann model. Central to the Von Neumann model is that you have but one memory, and store both your code and your data there. Notice carefully that a Java program (or C#, or whatever you like) is a manifestation of the Turing model. You set your program in concrete, once and for all. Then you hope you can deal with all data that gets thrown on it.
Lisp maintains the Von Neuman model; there is no sharp, pre-determined border between code and data. Programming in Lisp opens your mind to the power of the Von Neumann model. Programming in Lisp makes you see old concepts in a new light.
Finally, being interactive, you'll learn to interact with your programs as you develop them (as opposed to compile and run). This also change the way you program, and the way you view programming.
With this intro I can finally offer a reply to your question: Will you find places where it outshines "traditional" languages?
If you are an advanced programmer, you need advanced tools. And there is no tool more advanced than Lisp. Or, in other words: The answer is yes if your problems are hard. No otherwise.
In response to @lassevk:
One of the main uses for Lisp is in Artificial Intelligence. A friend of mine at college took a graduate AI course and for his main project he wrote a "Lights Out" solver in Lisp. Multiple versions of his program utilized slightly different AI routines and testing on 40 or so computers yielded some pretty neat results (I wish it was online somewhere for me to link to, but I don't think it is).
Two semesters ago I used Scheme (a language based on Lisp) to write an interactive program that simulated Abbott and Costello's "Who's on First" routine. Input from the user was matched against some pretty complicated data structures (resembling maps in other languages, but much more flexible) to choose what an appropriate response would be. I also wrote a routine to solve a 3x3 slide puzzle (an algorithm which could easily be extended to larger slide puzzles).
In summary, learning Lisp (or Scheme) may not yield many practical applications beyond AI but it is an extremely valuable learning experience, as many others have stated. Programming in a functional language like Lisp will also help you think recursively (if you've had trouble with recursion in other languages, this could be a great help).
complicated syntax??
The syntax for lisp is incredibly simple.
Killer app written in lisp: emacs. Lisp will allow you to extend emacs at will to do almost anything you can think of that an editor might do.
But, you should only learn lisp if you want to, and you may never get to use at work ever, but it is still awesome.
Also, I want to add: even if you find places where lisp will make sense, you will probably not convince anyone else that it should be used over java, c++, c#, python, ruby, etc.
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