Used to be the Amazon Music Downloader, now it's Amazon Music which handles Prime music, downloading MP3's you've bought and also accessing / uploading to your cloud-based music account, including any MP3's you have which were not purchased on Amazon. However, the music downloader is still available if you would prefer to use that.
I've never had any issues personally, I am a Prime member but was using their cloud services way before that. I've only ever bought one 'digital' (i.e. MP3) album, ever. If I really want some music I'll get the CD (if possible, auto-ripped on Amazon), then rip it and therefore have access to it more or less wherever I am.
I've not really had any problems with using Amazon's music services/software, other than the (new) Amazon Music software's rather unintuitive interface. However, they're hardly alone in providing software with a crap interface.