- máma -> mámo, mami
- táta -> táto, tati
The -o variant sounds more serious, reserved. As VladaFu pointed out, it's used more with Dad, probably because the relationship is less "tender" than with Mom. The -i variant is more familiar, casual, nicer.
There are other forms, "mamka -> mamko" and "taťka -> taťko" are pretty common, also quite familiar/tender.
There are many forms. Consider mum, mom, mummy... Mámo does exist but is not much used these days. You will hear it in fairy tales and older stories. Táto might be more common for some reason. The usual form is tati and mami.
Mámo may be used to call your wife in a humorous way. Maybe more often than to call your mother, these days.
I'd agree that something like, "Mom! Quick! Look out the window!" would not be at all unusual in the US. But a translation like that is heavy on exclamation points and, without the enthusiastic punctuation, I think it would fall flat. And probably generate a lot of comments from other learners that "No one would ever say that." :-)