Addition (Besides / Moreover)

Adding a second fact that makes the first more striking than it already was.

A。しかも B。 vs A。おまけに B。: Use しかも. It piles on in a way that says 'and on top of that, of all things', which works in speech and in writing. おまけに is casual and carries a grumbling or gleeful tone, so it flattens a serious point into an anecdote.

Appending a note to an official announcement or a written notice.

A。なおB。 vs A その上 B: Use なお. It introduces supplementary information in documents ('note that...'), which is the one job the others cannot do. その上 adds weight to the preceding point conversationally and reads as out of place in a formal notice.