I've been painting and printing some 3D terrain and as I look around for inspiration, I noticed that some terrain makers have chosen to put their buildings and other terrain pieces on bases, while others leave them bare back. Is there an advantage over adding a base or not?

One reason for not putting it on a base is so that it looks correct on different themed tables. You don't have a jungle base on a Hoth based table, etc. I usually base most of my terrain, but use a neutral brown or grey color scheme that wont stand out on most tables.

Base it for home games (your table). It looks so much better.

If you use a tall grass mat basing helps define areas in the grass.