Well, the deck is a bit stacked in the way you ask the question. You offer
X and Y are what drive them to do Z
X and Y are what drives them to do Z
In both examples, you have treated X and Y as a plural concept, as indicated by the use of are instead of is. In that case, what is logically plural and should take drive, the third person plural form.
However, there may be circumstances in which you want X and Y to be seen as a collective, a single driving force. In that case you could use a single verb form throughout
Fish and chips is what drives them into the restaurant.
But where the forces are independent (either alone might drive the actors), you should preserve the plural throughout
Both dedication and monetary gain are what drive people into the field of medicine.
The above answer was crafted before the illustrative sentences were added, but the logic is the same. It still is not clear as to whether both characteristics must be present for the result or either one alone might drive the result.
It seems evident that self-confidence and a desire for power are things that can drive them to enter this competition.
It seems evident that the concurrence of self-confidence and a desire for power is what drives them to enter this competition.