Adding to what's said - in a very un-specific, un-technical and non-rigorous manner that is entirely open to refutation and correction
the basic advantage is that everything is larger - from the line pairs per mm captured across the entire frame to the cameras and lenses to the 'normal' focal length... making it easier to get excellent resolution from the system. In this day of post-film photography, the cost is no longer much larger than 35mm.
You don't get as many pictures per roll of film, and therefore whether you process it yourself or send it out, the cost per image works out higher for 120 film.
The scanning advantage is with 120 film, though you need a scanner that will do 120. One reason is that everything is bigger on the bigger film. Another is that 120 film tends to lie flatter than 135.
But all that aside, in practice, say you have a 35mm camera and a 6x4.5 camera and you want to get a close in shot of a cat. The 35mm has a 50mm standard lens and the 645 has a 75mm standard lens, f/1.7 for the 50 and f/2.8 for the 75, say.
When you get close enough to fill the frame with cat head and paw in both viewfinders, at identical apertures the 645 camera will have shallower DOF than the 35mm camera. To equalize this you need to open up the 50mm lens or close down the 75.
With a 6x6, 6x7 or 6x9 camera the DOF difference is even more pronounced. And, as you know, with an APS-C camera you get more depth of field, and with a compact point & shoot you get even more...
This to me is the crux the matter: portraits using a 75 to 105mm lens framed to fill a 120 film frame can give a depth of field that naturally and gradually goes out of focus towards the back of the head (or tail, with an animal). This to me is an ultra cool and classic look. Same goes for other types of shooting.