We don't know where the fence is in relation to the property line.
He would have the right to lean over his fence and cut anything that extends over the property line.
True. If the fence is "shared" meaning the fence is exactly on the property line, however, the neighbor can cut jasmine that is growing on his side of the fence and what is growing through the fence but he cannot reach over to cut down any of the jasmine that is growing on sthoover's property. The neighbor can remove any plants that have rooted on his side of the fence. If the fence is entirely on the neighbor's property, on the other hand, the neighbor can cut down whatever is growing on the fence that he does not want growing there.
What might be a reasonable solution to the jasmine problem is for sthoover to install between the jasmine and the shared fence another "support fence" that the jasmine can cling to and grow on, keeping it off and away from the shared fence. This will not prevent the jasmine from rooting on the other side of the fence but, if it does, the neighbor can take care of whatever is growing on his property. Or sthoover can transplant the jasmine away from the property line entirely, resolving the dispute that way.
Although jasmine is fast growing and might be seen by the neighbor as invasive, encroaching as it does on his property, it does not appear to be on the Texas list of invasive plants that is regulated by the state.
http://www.texasinvasives.org/plant_database/