I'm always astonished when people conflate these two things. Lisp encourages you to put the forms that evaluate to a value in the position you would otherwise put a variable that has been set to that value, sortof code in place. That is kindof like functional programming. However, lisp does not avoid side effects or mutability, in fact virtually all operators in lisp are either destructive or have a destructive equivalent. The destructive equivalents are there because if you are limited to the nondestructive versions only you can't implement lots of algorithms without doing a bunch of consing and copying. In lots of ways it is easier to implement a performant functional language if it is pure and without side effects, because you can make strong assumptions everywhere.
Lisp was much more interesting 10 or 20 years ago than it is today. Today, most languages have stolen most of the good parts of Lisp. Strong but dynamic typing (or the gross weak but dynamic typing), pass by reference only, generic data structures, garbage collection, first class functions, ability to introspect objects at runtime, and most other features it pioneered are just taken for granted in modern languages. What is left as an advantage is the lisp syntax, and advantages that come from that, such as data-as-code and therefore powerful macro systems. That is also the main disadvantage to lisp, as much as people will claim that eventually you can see through it, I have never seen anyone do math either on paper or on a whiteboard in anything but infix notation. There are infix and postfix calculators, but I have never seen a prefix calculator.
TLDR; Lisp isn't really that functional unless you use a restrictive and inefficient subset of the language.