On this video, Ryan Connolly from Film Riot takes a look at both the visual and emotional differences between a zoom and a dolly shot.
Getting your camera closer (or moving away) from your subject can be done in different ways. You can physically move closer or away from your subject – in the film industry this has been done historically using a dolly although stabilizers of different kinds have been employed as well (and are becoming increasingly prevalent in recent years).
You can of course also zoom in (or out) – this is done in camera (by using a zoom lens) or in post production by digitally zooming in (although this will loose resolution in the process and is therefore less used in film productions although with modern ultra high resolution digital cameras – this can be quite acceptable for video work).
Connolly also has some tips like having a subject and a background when you moving into the frame and using a good follow focus (if you can spare the money a wireless one).
There are also some dramatic aspects to zooming or sliding in – it has to do with how the background looks in the process and you need to examine both options and see what works for your particular scene (Connolly gives some useful examples here).