With respect to garbage collection only and ignoring things like reliable debugging support, the primary thing it does is compaction.
If your memory manager does not compact the heap (i.e. never moves anything), then this implies a couple of things:
1. You can run out of memory whilst still technically having enough bytes available for a requested allocation, if those bytes are not contiguous. Most allocators bucket allocations by size to try and avoid the worst of this, but ultimately if you don't move things around it can always bite you.
2. The allocator has to go find a space for something when you request space. As the heap gets more and more fragmented this can slow down. If your collector is able to move objects then you can do things generationally which means allocation is effectively free (just bump a pointer).
In the JVM world there are two state of the art collectors, the open source G1 and Azul's commercial C4 collector (C4 == continuous compacting concurrent collector). Both can compact the heap concurrently. It is considered an important feature for reliability because otherwise big programs can get into a state where they can't stay up forever because eventually their heap gets so fragmented that they have to restart. Note that not all programs suffer from this. It depends a lot on how a program uses memory, the types of allocations they do, their predictability, etc. But if your program does start to suffer from heap fragmentation then oh boy, is it ever painful to fix.
The Go team have made a collector that does not move things. This means it can superficially look very good compared to other runtimes, but it's comparing apples to oranges: the collectors aren't doing the same amount of work.
The two JVM collectors have a few other tricks up their sleeves. G1 can deduplicate strings on the heap. If you have a string like "GET" or "index.html" 1000 times in your heap, G1 can rewrite the pointers so there's only a single copy instead. C4's stand-out feature is that your app doesn't pause for GC ever, all collection is done whilst the app is running, and Azul's custom JVM is tuned to keep all other pause times absolutely minimal as well. However IIRC it needs some kernel patches in order to do this, due to the unique stresses it places on the Linux VMM subsystem.