The basic difference between the two is emotions it appeals to.
Fun is enjoyable. It causes pleasure - especially in active forms, as thrill, exhilaration, challenge, elation.
Interesting appeals to curiosity - learning, it's about things we want to know, see, learn, examine. Whatever reasons - be it for pleasure, or e.g. for professional interest or satisfying anxiety.
Since usually satisfying curiosity is pleasurable, these two are often correlated, but not always. A gruesome sight, say, bowels sticking out of a live person's ruptured abdomen, may be interesting, but definitely not fun. A secret document on enemy military movements will be interesting too, but definitely not fun. The pilot of a damaged airplane will definitely find the damage report interesting and absolutely not fun.
On the other hand, if you solved a hundred crosswords, solving one more may still be fun, but hardly interesting. Riding down a slide in entertainment park will be described as fun too - it may be interesting the first time, but then you're not curious about it any more, you just do it for fun. Being given unexpected presents is fun - but since you didn't expect them, you didn't have time to find that interesting.
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.
Best Answer
Both sentences are grammatically correct and acceptable.
"In order to make no mistakes" implies that you have a target of zero mistakes, which isn't as strict as the latter.