I read an Adobe article which demonstrated H.264 video playback support in Flash Player. Would video encoded in this format be more compressed than typical FLV videos? Should I be using this format for video I place on websites from now on?
The FLV files contains video encoded with various codecs: VP6, H.263, H.264. I suppose that by typical you are referring to the VP6 encoding? If yes maybe this article can help you http://support.on2.com/h264_faq.php.
H.264 is one of the most advanced high quality CODECs out there. The streaming media industry is generally moving that way, including Adobe who is the author of FLV.
This move to H.264 is saving folks money in bandwidth, and/or allowing them to increase quality. So...
Yes, you should be using h.264 instead of H.263 or VP6, which are in FLV files.
To contradict Cornel Creanga's answer, partially, placing H.264 in FLV files in not recommended by Adobe. While technically possible, it is uncommon, problematic and the FLV container cannot support all of the advanced features that h.264 uses to work it's magic. Don't try it.