Are there two different versions of the \mathcal{F} character?

The final version of Computer Modern's mathcal F is the second one. To see it, you can compile with metafont the file cmsy10.mf (located in [texmf]/fonts/source/public/cm/; be sure you have the latest version). Here is the result, after conversion in dvi format with gftodvi:

If you compare the type 1 versions of Computer Modern, you see that the Bluesky/AMS and Latin Modern versions both have the right version. However, old versions of Latin Modern have the wrong mathcal F:

This means that if you use a recent TeX distribution, you should always have the final version ofthe mathcal F.

As an aside, my version of Computer Modern Typefaces, although recent, still shows the old version:

If you include the command \usepackage{eucal}, then \mathcal{F} will produce the curlier one and \CMcal{F} will produce the other. It's up to you to decide which you prefer.

To more directly answer your question: Including the command \usepackage{eucal} changes the \mathcal command so that it will produce the curlier one; omitting that package will make \mathcal produce the less curly one.