'Autonomous' means 'self-directed'. Auto - nomy. From the Greek 'autos' - self, and 'nomos' - law. It means that your drive to act comes from inside yourself.
'Independent' means 'not influenced by outside forces'. It is from the french 'in' - not, and 'dependant' - hanging from. It means 'not hanging from' - or 'not dependent on' anything.
So although the meaning is similar, it is different, as you say.
He is completely autonomous as a freelancer and defines his own programme.
The child is able to play autonomously - she makes up her own games.
The freelancer is independent of any company - no-one tells him what to do.
The child is able to play independently - without her parents' supervision.
Comment: it's always really interesting, I feel, to look up the etymology of words when wanting to understand them. The etymology depicts their origin, as well as the constructing words they are made from - and is often more accurate and easier to understand than reading the 'definitions' of the words - which are often more like 'interpretations of the words, in use' rather than being the very original meaning. The etymology goes more to the source and often reveals the history of the word as well, which the word will often carry, as part of its cultural meaning, and this I think is so useful and rich, and clear, when writing and expressing oneself.
Lastly, to come back to your question example - 'is autonomous and independent' means:
- is self-directed and not influenced by others