Additionally is seen as sloppy and informal as compared to "in addition". Its often used(in poor taste) as a replacement for "furthermore". Consider the sentences below:
Additionally, I would like to propose a toast.
In addition, I would like to propose a toast.
Furthermore, I would like to propose a toast.
All sentences are valid, but the first is most sloppy, the second is in the middle, and the last is the best received of the three.
But given a sentence like this:
- I would like a faster car, in addition to a surround sound system.
In addition makes the most sense here.
The word "Additionally" should be avoided in nearly all situations if you want to be formal or sound more professional.
Response to Edit:
More than that, is fine but is somewhat verbose. Furthermore is your best bet.
Source: Im a native English speaker raised in the US, and this source backs up my beliefs: