Also, remember, you are not the center of the universe. (I am 
) She may not be avoiding you, she might just be busy.
Just be nice when you do see her. If she seems cool, then apologize. If not, let it go.
Has she ever been noisy at odd hours before or was this a one-time incident? If it was a one-time thing, you need to eat some humble pie. We all do things that inconvenience our neighbors occasionally.
Also, if she is at all friendly with your other neighbors, you need to apologize soon or else run the risk of becoming known as “that crazy person who hollered at nice Mrs. So&so.”
Apologise. And offer assistance in cutting wood. You can’t buy good neighbors, and having a local feud can be soul destroying.
Another vote for an apology. I think you’re overfocused on the fact that you guys have little in common (at least superficially), and for some reason that’s been holding you up. If anything, those differences should be prompting you to get off your duff and patch things up even moreso, since tensions between people who view each other as outsiders can easily get out of hand.
Saying sorry will make you feel better.