Moving in will change the relative perspective of objects. Things that are a little closer will appear disproportionately bigger, whereas from further back this size difference due to distance difference is reduced. A great example is getting close to someone's face, which causes the subject's nose to appear unusually large. Other times, that perspective difference is exactly what you want.
Zooming in has the advantage that it can be done from the same spot, and you don't actually have to get close to the subject. A obvious circumstance where this is important is when taking pictures of wild animals. They will run off or fly off, or in some cases try to eat you if you get too close.
Moving in allows a shorter focal length to be used, which allows for a slower shutter speed at the same level of blur due to camera wobble, like when hand holding. Moving too far in may not work with your lens. All lenses have minimum focus distances. If you really need to get up close, then you want a macro lens, extension tubes, closeup lenses, or a reverse mount.
Everything is a tradeoff.