There is an example in another thread here which has the following example: We spontaneously decided to get married. I would translate that into: Wir haben uns spontan (dazu) entschieden zu heiraten.
What is the difference in these two? Same words, same meaning to me as a native German speaker. As I see it, spontaneously can have a subtle other meaning than spontan, but can also be the same.
But we can resolve this confusion around "spontaneously" by going to its actual origin: The English word "spontaneous" is a loan word and originates from the Latin "spontaneus" which itself derives from the Latin noun "spons" (feminine) primarily meaning "free will". It also has a secondary connotation meaning "impulse". [1]
So the common English usage is closer to how the Romans used the word. That does make sense: The Romans conquered the British isles but failed to win against the Barbarians that lived east of the river Rhine. The Germans only much later incorporated Latin vocabulary and did so mostly only in elite circles. That's possibly why "spontan" means "acting out of impulse".