I always thought that these 2 are the same. But since you started this topic, I had to look in google for it's different. I won't include the definition in this post but I will try to explain it by example based on my full understanding here.

Referral: Your referral link which you used mostly to send to your friends, relatives, etc. Yourself if you're not a streamer, vlogger, etc.
Affiliate: It is bigger type. It is used widely by streamers, vloggers, etc. wherein they advertise the site at the same time of gaining commissions from everyone who registered using their referral link.