"Handy" implies usefulness.
"convenient" implies close at hand.
A tool box is not necessarily useful, just because it is close at hand. You have to know how to use the tools.
There are contexts where the two words are similar in meaning or at least the differences in meaning do not conflict.
The store is handy as it is close by and open 24 hours a day.
The store is convenient as it is close by and open 24 hours a day.
The above sentences convey the same general meaning.
However, the following do not:
He is very handy with a hammer.
He is very convenient with a hammer. ("convenient" makes little sense here)
So it depends on context.