Something that is tantamount is equivalent to something else. For example, we could say that a pitcher of beer is tantamount to a six-pack.
Something that is paramount is (1) of chief concern or (2) supreme in rank. The word is often used in somewhat redundant phrases such as of paramount importance. This is redundant because something that is paramount of importance is paramount, period. This usage is widespread, though, and isn’t wrong.
Examples
Tantamount
If people consider terminating pregnancy to be tantamount to murder, they will have little tolerance for the differing beliefs of others. [Los Angeles Times]
But money given freely isn’t tantamount to free money, regardless of the method. [National Post]
Paramount
The only true justice is open justice, yet increasingly judges in the UK see the right to secrecy as paramount. [Independent]
Erica said in an interview last year that the individual personalities of her children were paramount when it came to choosing schools. [Sydney Morning Herald]