Well, it depends what you mean by ''crappy weather''. If you mean cold, yes it is cold February.
However, in Tokyo (and along the Pacific coast) the weather is usually mostly clear, sunny and dry. I'd say that 5 days a week it is just clear and sunny. On the other hand, the Sea of Japan coast is often cold and snowy. In fact, many people come to Japan in February to go skiing, as there are literally hundreds of places you can ski. And the winter months are about the only ones when you can see Mt. Fuji almost every day.
On the other hand, I'd say the months of June, July, August, September are the worst, with hot humid and wet conditions.
As for the original question most people would find March better, as it's not as cold, still fairly clear and sunny, and the blossoms are just starting to appear.