It isn't quite that simple.
"the words "begin" and "start" are basically synonyms, although there
are some differences in usage. On the other hand, while both "end" and
"stop" are antonyms of "begin" and "stop" they don't have quite the same
meaning, i.e. they aren't synonyms.
The terms "start date" and "end date" are idiomatic combinations that
are accepted in usage, whereas "begin date" and "stop date" are
unacceptable usage.
Languages aren't strictly logical.
Regards, Einde O'Callaghan