I often get asked how to pronounce these two words. Are they the same or different?

They are different. Here is how to pronounce them:

For “worm” say “were” and then add a “m” sound at the end.

for “warm” say “wore” (past tense of wear) and then add an “m” at the end.

So “warm” actually has an “o” sound in it, but “worm” has an “er” sound like the words “her” or ” “bird.”