Swings and roundabouts...

Moving bed is bad if the part you are machining is heavy, or if you've got a heavy vice etc as the steppers have to accelerate that mass quickly. If however the mass of what your machining+bed+clamps is less than the gantry then moving the bed will get better acceleration/speed.

How rigid the machine is depends on how you configure either option, so it's difficult to compare. Having the gantry stationary is good in that you can make it as strong and heavy as you want want ...

Not sure what other pro's/con's there are as I never really considered it ... other than space as you say.