wet spots on inside of garage wall
I see several problems:

1) Moisture in the joints not the cells:  If you look closely at the pictures, you'll see that moisture is penetrating at the mortar joints (vertical and horizontal). That means the cells are probably grouted solid (that's good), but they probably didn't grout the vertical joints (the space between the blocks). Masonry is VERY POURUS. When water penetrates the "face shell" of the masonry, it probably finds a void between the blocks and then runs down the wall and fills up the void or "wicks" through the wall an migrates into the garage. (I know, I saw the paint on the exterior too, but it's useless keeping water out, unless you use a "block filler" first. Once water is in, the paint is probably enough resistance to keep it from coming out...it's just easier to migrate towards the warmer side.)

2) Mortar joints:  Your close-up picture shows some of the mortar joints (vertical and horizontal) are not "struck" or "tooled". Striking or tooling does two things: a) compresses the mortar so it's more waterproof, and b) removes any extra mortar that may create a lip to catch water running down the wall. 

3) Your front wall (or back wall...the one with the sledge hanging up) has the same issue, but minor. This would indicate that it's not getting the same amount of water on it. So, that means your neighbor's roof is still causing problems. Yes, the gutter was great, but not all water carefully goes into the gutter in a storm.

4) Your masonry wall extends about 3' above Your neighbor's gutter. From the inside, you can see that the "wet spots" start 2' down. The wall above the gutter seems dry inside...probably due to the drying effect and partly because it doesn't have enough water in the wall to fill up to that point. 

Repair??  Hmmm...it's too narrow to work in, so I'd experiment with hanging a tarp in the gap. If it decreases the moisture spots, then you need to fill and paint the wall. (I'll loan you my 9 year old grandson, who is skinny enough to get in there...and still be able to get out.) 

You could also experiment with drilling 1/4" holes near the bottom of the wall in a couple of the wettest spots to see if water runs out. It would be like a weep hole...but make sure water has a place to go, so it doesn't back up and flow into the hole. (Be prepared to fill the hole if things go bad.)