Not really. 'I have to go', is more of a command or something absolute. (e.g. I have to go to the bathroom) 'I have to get going' , suggest it is a future event (e.g. I have to get going soon) The similarity is when you use the term, ' I have to get going now'. This is the same thing as saying ' I have to go'. The other difference is you would not say 'I have to get going to the bathroom'

"I have to go" is more urgent and less formal, while "I have to get going" is a bit softer. If you are on a phone call with someone you don't know well, and you need to hang up, "I have to get going" is a better choice than "I have to go." It's a subtle difference.