What is the difference between <strong> and <em> tags?

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.

<strong> is a tag you'd put around a sentence or phrase to indicate that "this is more important than the surrounding text".

<em> is generally used to indicate the stress of a word within a sentence.

For example:

In spite of what some might say, there is a semantic difference between the elements.

For a clear distinction between stress emphasis and importance, and more examples, see the HTML 5 draft.

In HTML4:

EM indicates emphasis.

STRONG: Indicates stronger emphasis.

http://www.w3.org/TR/html4/struct/text.html#h-9.2.1

Note that the difference is going to change:

In HTML 4.01, the <strong> tag defined strong emphasized text, but in HTML 5 it defines important text.

Doesn't make the issue any clearer, does it?