what i meant about player is the broad meaning of "player" which means the object in question that would "play" the video, some people use a cellphone, others use a video player on a literal sense. although you're using a PC, its not a guarantee that you'd be able to support full 1080p without the machine's specifications, for example an old pentium dual core would suffer decoding hi10p 1080p and so on.
with a monitor not having a resolution of 1080p it wouldn't be as nice when playing 1080p videos on it, since it's gonna get downscaled back to whatever the monitor's resolution is, although some results to a sharper image but still its not a good idea.
but based on some of what you've said, i can say "go ahead" and grab 1080p if its available, otherwise you can just grab 720p, theres really no point of arguing 720p vs 1080p without a reason of "why not".
without a "why not" its always 720p < 1080p.