Language Peeves - "I've got to" vs "I have to"

From a reader -

I would like to hear your take on people saying, "I've got to..." rather than "I have to..."

The comparison here is between two phrases with the modal verb "have" that are almost identical semantically.

Both "I have got to" and "I have to" are used to indicate an obligation that has been imposed upon the speaker by another person (if the speaker was talking about a self-imposed obligation, the modal "must" would be used).

Both phrases are grammatically correct and the only real difference is the formality. "I have got to" is generally considered less formal than "I have to."