Functional languages, by definition, should not maintain state variables. Why, then, do Haskell, Clojure, and others provide software transactional memory (STM) implementations? Is there a conflict between two approaches?
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There is nothing wrong with a functional language maintaining mutable state. Even "pure" functional languages such as Haskell need to maintain state in order to interact with the real world. "Impure" functional languages like Clojure allow side effects which can include mutating state.
The main point is that functional languages discourage mutable state unless you really need it. The general style is to program using pure functions and immutable data, and only interact with "impure" mutable state in the specific parts of your code that require it. That way, you can keep the rest of your code base "pure".
I think there are several reasons why STM is more common in functional languages:
I personally like Clojure's approach of allowing mutability, but only in the context of strictly controlled "managed references" that may participate in STM transactions. Everything else in the language is "purely functional".
Note the above code is fully transactional and atomic - an external observer reading the two balances within another transaction will always see a consistent atomic state, i.e. the two balances will always sum to 200. With lock-based concurrency, this is a surprisingly hard problem to solve in a large complex system with many transactional entities.
For some extra enlightenment, Rich Hickey does an excellent job of explaining Clojure's STM in this video
Sometimes a program requires mutable state (for instance, database contents for a web app) and it would be great to be able to use it without losing the benefits of functional programming. In non-functional languages, mutable state permeates everything. If you make it explicit with some kind of special API, then you can confine it to a small identifiable region while everything else remains purely functional. Benefits of FP include easier debugging, repeatable unit testing, painless concurrency, and multicore/GPU friendliness.
Your definition is wrong. Language that cannot maintain state simply cannot be used.
The difference between functional and imperative languages is not that one of them has state and the other does not. It's in a way they maintain the state.
Imperative languages have state spread all over the program.
Functional languages isolate and maintain the state explicitly via type signatures. And that's the reason they provide sophisticated state management mechanisms like STM.