Both of them emphasize text. The <em> tag shows text as italics, whereas <strong> makes it bold. Is this the only difference?
Yeah, the definition of what ‘strong emphasis’ is compared to just ‘emphasis’ is pretty woolly. The only standard definition would be “it's emphasised, but more!!”.
Personally I use <em> for normal emphasis where you'd read the emphasised word in a different tone of voice, and <strong> for that thing where you take key words and phrases to pick them out of the text to help people skimming the text pick out the subjects.
This isn't a standard interpretation, but it makes some sense and rendered appropriately by the default italic/bold styles. Whatever you do, be consistent.