Do men have fewer ribs than women?

NO!!

Both men and women have twelve ribs on each side, making 24 in total.

The ribs, and the musculature in between them, are an ingenious compromise between making the 'shield' for the heart and lungs as strong as possible, and still allowing the chest to expand when we breathe.

You can see in the diagram that the first 10 ribs are connected via a cartilage strip to the sternum* (breast bone), whilst the last 2 aren't (they go by the evocative term 'floating ribs'). Your 12th (last) rib actually ends quite a lot more laterally than you'd imagine. You can feel it by running your hand across the bottom of the ribs, starting from the centre. Suddenly, at your flanks, you feel an 'extra' lump = the 12th rib.

* Confusingly, the 10th rib's cartilage isn't properly connected in this picture. This can occur, but it's the exception, not the rule. Unfortunately, it was still the best picture of the anatomy I could find...