I am happy to be going home. I am happy to go home.
I use both for a same meaning, but I was wondering if there is any difference.
There is a difference, yes.
I am happy to go home states that you are pleased with the notion of going home, whenever it might be. The concept of going home is favorable to you and the feeling is consistent.
Q: Is it depressing when the school term ends?
A: Maybe a little. But I am happy to go home.
As opposed to I am happy to be going home, which implies that said behavior is imminent. The action is underway and in progress. You're in the process of returning to your place of residence.
Q: Are you excited that you're leaving today?
A: Yes, very much. I'm happy to be going home.