“J’aime beaucoup” is stronger than “j’aime bien”.

As stated by oli, one has a stronger meaning than the other.

“J’aime beaucoup ” can be translated by “I like very much” whereas “J’aime bien ” will be translated by “I like”.

“J’adore ” is an even stronger form than “J’aime beaucoup ” and will be translated by “I love” (without love-between-two-persons connotation).