Big difference, actually. And it's not because one has an 'm' while the other sports an 'n'. LOL.

EM DASH: is roughly the size of an 'm', and sometimes called a double hyphen (--). It is used to indicate an added emphasis, interruption, or abrupt change of thought. It is also used (sparingly, I hope) to replace colons, semicolons, commas, and parentheses.

e.g. I like you—really like you—for being a good friend. [added emphasis]

      She dashed off to go to the party—but wait, she forgot her purse. [interruption]

      "I know, but—" The baron picked up his gloves. "—this plan of yours is sure to fail."  [interruption]

      I wish you would—oh, never mind. [abrupt change of thought]

In most Word documents, you can create an em dash by making a double hyphen (--) and pressing ENTER. Word automatically converts it for you. Also, there are no spaces in between the word prior to the em dash, and the word after it.

THIS: party—but
NOT: party — but

Em dashes also precede quotation attributions.

e.g. A professional writer is an amateur who didn't quit.
      —Richard Bach
    

EN DASH: is traditionally half the size of the em dash, or is roughly the size of an 'n'. It is technically wider than a hyphen (-). They're used to:

1. Connect a range (such as numbers).

e.g. 1–10
       pp. 35–40

Non-numerical range:

e.g. January–December

2.To contrast values, or illustrate a relationship between two things.

e.g. Father–daughter event
      Boston–Hartford route

3. When combining open compounds:

e.g. a high school–college conference
     

These are the simple definitions—we could probably get even more technical and discuss other aspects of the em dashes and en dashes, but unless you're writing formally, these will do.

You can find more about this topic here and here and here. Also, I found a wonderful blogpost by M.E. Summer of Sticking to the Story about hyphens. Check it out here.

Questions? Comments? Any info you'd like to share? Please use the comment section. Thanks!