This is a great question, and something native don't usually think about. :)
It seems to me that "Intense" is used more broadly and more commonly, and that"Drastic" is often used to describe extreme changes or methods.
- That was an intense ride.
- She looked at me with an intense expression.
- He made drastic changes to the report.
- After the usual medication didn't work, the doctor recommended a more drastic treatment.
I hope this helps!