One difference is the former is very rarely used while the second one is very common.
The main and most subtle one is the difference in agreement.
In the first sentence, “au doigt” is an optional information, the verb “se couper” is reflexive, “coupée” agrees with its feminine direct object s’ (herself), i.e. “elle” because it is located before the verb.
Elle s’est coupée au doigt.
In the second sentence, “se couper” is still reflexive but here, “le doigt” is no more optional but the direct object of the verb while s’ is an indirect object (to herself). The direct object is located after the verb so there is no agreement.
Elle s’est coupé le doigt.
A third difference is the first sentence can only be used when she only superficially cut her finger while in the second case, it can also be used in the rare cases the finger has been totally severed.