I'd be putting up a fence.
Our backyard also backs up to the woods, dh built a three foot high fence with wooden cedar pickets. The pickets are 6 inches wide and he left three inches between them. Since the fence is made of the same material as the privacy fencse on the sides, I think it looks really nice. In fact our next door neighbor "copied" us
I wonder if your neighbor will just let himself in to weed eat the back side of his fence? I can't imagine that happening in my neighborhood. My cutting through the yard for kids, but never adults.