Bottom round is the traditional name in United States. Outside that tradition, butchers carry comparable beef cuts under different names — sometimes the same anatomical piece, sometimes a close cousin. The alternatives below are grouped by country so you can match what your local butcher actually carries.
Both are part of the wider outside-round muscle group anatomically; eye of round is the small, cylindrical eye muscle within the silverside group, bottom round is the larger surrounding mass. UK butchery often groups them all under "silverside."
Both lean round subprimals. Bottom round (silverside) is the harder outer-thigh group; sirloin tip (knuckle) is the front-thigh quadriceps and is naturally tenderer.