Menstrual discs sit above the vaginal canal in the widest part of the vagina, called the vaginal fornix. Cups sit in the canal, similar to where a tampon sits.
Both the disc and the cup can be worn for up to 12 hours, but what you do once you take it out is another point of difference. The disc is disposable, so you simply empty the contents and throw it away. The cup can be reused, so after emptying the contents, you clean it with mild soap and water and reinsert.
Here's a comparison chart: