Those are burr knots. Basically root initials. If you buried those limbs they would quickly root at those spots. The woolly aphids like any crevice they can find and and hide/feed in those spots.

The tree is OK if it's fruit suits you.

I've had a lot of crown gall on stone fruit but haven't seen it yet on apple. So apple may be more resistant.

Thank you so much. I've only read about crown gall and haven't experienced it in person. I'm posting a couple more photos just to be sure. It seems that the "growths" are growing on old pruning wounds, which is one reason why I was worried it was crown gall. But hoping you are right! I'd hate to lose it.

Another picture on a smaller branch.
Thanks again!

They sure do look like burr knots, although I don't recall seeing them so high up in a tree
I have seen similar growths on black locust, I believe those are a virally induced burls. It could be something similar in your case.
Either way I don't think they hurt the tree unless they become numerous enough to "girdle" the tree.
On locust they will weaken the tree structurally. I would assume that to be true for burr knots on apple as we'll. the chaotic grain is inherently weak.

Thank you all. I'm so glad to find out this tree is not diseased. It produces a lot of fruit and like I said, it would be sad to lose it. I'll keep an eye on it and keep it maintained. Thank you again gardenweb community!