That's tough. I've endured both terribly and both are infact, terrible. Honestly, I think it depends how bad the sunburn is. I have a really white complexion, so if I get a bad burn, it's just relentless sting, which I can't handle.
As far as strep, that's also horrible. It's like an enhanced sore throat that pains you times ten.
But if I had to pick, I'd go with strep. Why? You can drench yourself in cough drops or other sore throat remedies to help ease the pain. If I had a full body bad sunburn, which lasted for days, It'd hinder everything I'd be able to do physically. And that's no bueno.
Sunburn. I have not ever had a bad case of either but I feel like with strep throat you may have that overall-feeling-absolutely-awful feeling that comes with being sick whereas with sunburn it'd be more of just pain & lots of it.
I have had strep and couldn't sleep for a long time I went about 30 hours without sleep and I was not a fun person to be around, which I guess 30 hours isn't terrible but when you think of the fact I felt like crap on top of that it changes a bit. I have never had a terrible sunburn only smallish ones but my dad had one that covered his whole back and he said it got a lot worse before it got better... so that is a toughy I would guess strep because my dad has a pretty good pain threshold so when he says something hurts really bad it would probably be agonizing for me (I have a much lower threshold).