In which cases would you use which? Is there much of a difference? Which I typically used by persistence engines to store booleans?
BIT should only allow 0 and 1 (and NULL, if the field is not defined as NOT NULL). TINYINT(1) allows any value that can be stored in a single byte, -128..127 or 0..255 depending on whether or not it's unsigned (the 1 shows that you intend to only use a single digit, but it does not prevent you from storing a larger value).
For versions older than 5.0.3, BIT is interpreted as TINYINT(1), so there's no difference there.
BIT has a "this is a boolean" semantic, and some apps will consider TINYINT(1) the same way (due to the way MySQL used to treat it), so apps may format the column as a check box if they check the type and decide upon a format based on that.