Hmmm ... great question. Though there can be some overlap, I'd suggest thinking of excessively as meaning "way too much." For example, she laughed excessively at the party; she laughed way too much. Or there was an excessive amount of food at the banquet. There was way too much food. Think of "exceedingly" as "extremely." The food at the banquet was extremely (or exceedingly) good. To combine both: we ate excessively at the party (we ate way too much) because the food was exceedingly (extremely) good.

exceedingly = to a unusually great extent; very excessively = more than what is necessary, usual, or specified

Exceedingly means extremely. For example, it is an exceedingly difficult problem. Excessively means something is over a proper limit. Drinking alcohol excessively is dangerous.