Main benefit is you don't need to remember to start your editor frame in screen/tmux and you can use both terminal and gui/x11. So when I'm at work I start emacs in X11 windows, when remote I simply ssh in and do emacsclient and connect to all the buffers I have open (if I have low latency i'll do x11 forwarding and start x11 emacsclient ) It's basically the same as tmux/screen but gives you a lot more flexibility.

Other big benefit is starting emacsclient is basically instant, so opening a file from a webbrowser or whatever is super fast (some folks have 30 second or longer emacs startup times).

Only real downside is it really sucks bringing back your state when emacs crashes and you have hundreds of buffers open for multiple projects across 20-30 emacs clients.