See documentation
List.Exists (Object method)
Determines whether the List(T) contains elements that match the conditions defined by the specified predicate.
This exists since .NET 2.0, so before LINQ. Meant to be used with the Predicate delegate, but lambda expressions are backward compatible. Also, just List has this (not even IList)
IEnumerable.Any (Extension method)
Determines whether any element of a sequence satisfies a condition.
This is new in .NET 3.5 and uses Func(TSource, bool) as argument, so this was intended to be used with lambda expressions and LINQ.
In behaviour, these are identical.