The blocks where a vine is rooted to are impossible to break unless you cut the vine first. I wasn't sure if this was a bug, so I am instead suggesting that the more realistic approach be taken, which is if you break the block from which the vine dangles, down goes the vine itself. If this is a bug, this post should probably be moved.
I know, but if I break the block holding the root, it should come down entirely. The vine's root shouldn't make the block it is attached to invincible. I was mentioning that it is not only unrealistic, but it also creates the useless need to switch to an axe underground to cut a single vine root just to go through some path that specific block is blocking.