Yes, you should be fine with installing the new drivers, especially if you use the download utility from AMD's website (so that it auto-detects your GPU & the appropriate driver).
The only time you'll want to consider uninstalling old drivers is if you're physically replacing the GPU. And even then, that's only if there's a big gap between the new & old GPUs (i.e. replacing an HD 4850 with an R9 290X). Of course, too much of a hardware gap means you're probably replacing the motherboard and/or CPU anyway, which means a fresh install of Windows (& therefore fresh driver install).