In order to find the differences in use cases for the two, we need to take a look at how they can be used.
In combination with 'ein' to mark a small amount to mark a small amount of something. Here it is synonym to "wenig" and it's probably a matter of style which one you and your peers prefer to use.
Ich habe ein bisschen Geld. | Ich habe ein wenig Geld. = I have a small amount of money.
In combination with 'kein + substantive' to mark the fact there is nothing there of said substantive. Here 'wenig' can't be used.
Ich habe kein bisschen Geld. = I have no money at all.
In combination with "das ... substantive" to mark something as not really important. Aswell, 'wenig' can't be used here.
Das bisschen Geld nützt mir nichts. = That small sum of money is not helpful to me.
Other than 'bisschen' 'wenig is not only an indefinite pronoun, but can also be used as an adjective.
Das wenige Geld, das Onkel Karl verdiente, kaufte uns die wenigen Möbel, die wir besaßen.
So, if you need the comparative or superlative you can't use 'bisschen' but 'wenig'
Maybe, take a look at the wiktionary pages for bisschen and wenig for further examples and references.