How is the the process of multiplying fractions different from the process of adding fractions?

When adding fractions, you need to have a common denominator. Once you have converted the fractions so they have a common denominator, you just add the numerators together, and the denominator stays the same (don't add those together)
Example:
1/2 + 3/8   I would change the 1/2 into 8ths, so I multiply both numerator and denominator by 4 to get 4/8
4/8 + 3/8 = 7/8   I just add the numerators together, the denominator stays as it was, so not add those.

When multiplying fractions, which is easier, in my opinion, you just multiply the numerators together and multiply the denominators.
Example:
1/2 · 3/8 = (1·3)/(2·8)=3/8

I hope this helps