This is a good question, because currently is somewhat elusive to define, as you have no doubt seen by googling for it, and right now is only subtly different.
Meaning:
Currently refers to something that is true now, and in an ongoing sense, but is not expected to be permanent.
Currently we are having a hot spell.
... the weather is hot, but eventually it will change.
I'm currently not at my phone.
... and later I will be back and will listen to your message
Right now focusses more on the exact moment, and has more of an implication and expectation of change.
Right now it is too hot to go outside
... but soon, in the evening it will be fine.
I can't talk right now
... I'll get back to you as soon as I can.
(Note: later answers have correctly pointed out that "right now" can bring an element of imperative that "currently" does not).
Grammar:
There is a grammatical difference between the two.
Currently is an adverb. It describes something taking place (verb).
Right now is a noun - it is a time.
So they can be used interchangeably in sentences, but not without slightly changing the sentence (usually) to accommodate this difference.