I discovered the best combo by accident.
I had put black foam tubes over my drop handlebars, but it was getting a tad damaged in places.
I was rummaging around in my box of bits & bobs one day and found some padded handlebar tape. Only having a few minutes to spare, I thought I'd quickly wind it on over the foam already on the handlebars.
It was superb!
If you have large hands it makes the bars perfectly sized and really comfy.
So, I recommend foam tubes first, and then padded bar tape (self-adhesive) wound over the foam.