mögen means like

gerne can have different meanings eg:

willingly, gladly, with pleasure etc. like is only one of the possible meanings.

you can also use gerne to put a stronger emphasis on some words

eg :mögen or wollen. etc.

ich mag das - i like that

ich mag das gerne - i really like it

ich mag das sehr gerne- i really like it very much

ich will das- i want that

ich will das gerne- i really want it

ich will das sehr gerne- i really want it very much

we also use gerne as a respond when someone says danke (thanks) either just gerne or gern geschehen.