How do I limit the water flowing through my neighbor's fence into my yard?
You need to be careful how much you impede the natural flow of water off your neighbor's yard.  If your property is downhill of your neighbor's, the water flows naturally from his property to yours.  You could find yourself in legal jeopardy if you dam up or otherwise prevent that natural flow.

On the other side of the coin, your up-hill neighbor can not, and should not make changes to the topography of his property that would increase the amount of runoff onto your property. 

There a legal doctrine involving this kind of thing, but I can't remember what it's called.  Here's one reference I found on line.

https://www.hg.org/legal-articles/what-can-i-do-about-water-drainage-on-my-property-caused-by-the-adjourning-property-43279