Trick to Remember the Difference
If you are writing in modern times, you should choose emphasize, whether you are an American or British writer. This version is standard in both communities. Emphasise used to be more common in British English, but that appears to have changed around 1920.
Whats does Emphasizing mean?
to give emphasis to; lay stress upon; stress: to emphasize a point; to emphasize the eyes with mascara.
What does it mean when someone emphasizes something?
: to give special attention or importance to He emphasized the word "maybe."