Well, for one thing, the posts put up by the neighbour are suitable for post and rail and may be quite unsuitable for stock fencing. Stock fencing requires tension (and if not tensioned, is dangerous to sheep as well as to horses), so the posts are usually longer (so bashed in deeper), heftier, with much bigger and heftier strainers, etc.
It is entirely possible that the attaching of stock fencing to the posts of the post and rail fence will cause the latter to have a reduced lifespan. As it’s the neighbouri’s fence I think she’s entirely within her rights to object to object to the attachment of a second fence to her posts. And it would have been common courtesy to discuss it with her beforehand, in any case, so the lack of that courtesy has probably not improved her mood.
x-posted with Black Sheep.