Your Name, or Kimi No Na Wa, is an absolutely wonderful new Japanese animated film written and directed by the talented Makoto Shinkai. If you have a chance to go see it in theaters or watch it on Bluray or any other way, I would encourage anyone to do so — it is a visually gorgeous movie with great story and characters. It's beautiful, funny, thrilling, gut-wrenching and tear-jerking in all the best ways. It's probably in my top three movies of the year. This manga adaptation, though? I really don't see the point.
The book follows the film's story almost beat for beat. It doesn't expand much on anything, doesn't show any new perspective, nothing like that. The manga also looks inferior to the movie, because it's in black and white and is drawn by a single artist, as opposed to a whole big budget studio of professional animators. The story is still good, of course, but you only get about one third of it in this first volume, and the following books aren't even released yet. The sad truth is, this adaptation is completely unnecessary — watching the anime is a much better experience in every single way.
So the choice is easy: definitely watch Your Name. It's an incredible film. Skip the manga.