There are many differences between menstrual cups and discs, the main differences include: 

• Menstrual discs placement in the vaginal fornix leaves the vaginal canal free of obstruction and allows a person to have penetrative and oral sex while wearing it. Because a menstrual cup sits below the cervix, opening up inside the vagina and sealing around the vaginal walls it will need to be removed for sex. 
• Menstrual discs are available in disposable and non-disposable options, whereas menstrual cups are fully reusable. 
• Menstrual discs are smaller than cups and will likely need to be emptied more often for those with a heavier flow.
• Menstrual cups have been around for many years and were first invented in the 1930s. This means there are lots more options available in a range of shapes and sizes, so you will be able to find one that suits you. 

More info at the woman & home website.