Jython has been around a long time and is based on a mature language, Python. However, its development has stalled in recent years. Groovy is a relatively new language and thus still is developing.
Jython has two main advantages. First, the language is Python and has all of its strengths, including clean and consistent syntax, introspection, dynamic creation of classes, properties and attributes. Second, it runs at the speed of the JVM.
Groovy is a new language based on features from Ruby, Python and Haskell. However, its runtime environment is any JVM. The language syntax attempts to be Java-like, while making the form of the language much simpler. Groovy is a statically typed language, so it requires a compile step before producing Java byte code. Groovy does not currently have an interpreter, although it does have a shell.