Yes, drive speed is a factor, as your media has to go through a pipeline and then get written to the disk.  The type of connection matters.  eSATA is one of the faster external drive interfaces.  USB3 is even faster.  You'll see a severe bottleneck with USB 1 or 2. 

You can likely find a disk speed tester on the web that will tell you how many Mb/s your drive is reading and writing.  And you can probably get the spex for how fast the drive should be from the manufacturer's web site.

FAT-32 isn't a good format for video, because it has a 2GB file size limit.  I'd reformat it to NTFS (after you back up the files on it), which doesn't have a limit.  But, I don't think that will affect the write speed.  You may just have a bad cable.

Jay, if you charge by the hour, and won't miss a deadline, no big deal. 

However, if your exported file gets over 2 GB, expect to see an error message once that happens.

I do have to respectfully disagree with my esteemed colleage, lasvideo on this one (Sorry!).  I've often worked from and written to the same external drive, although in my case it was usually a FW400 or 800 drive, and the video was DV or XDCam EX.  eSATA is a faster protocol than Firewire.  MP4 isn't transfer intensive, as compared to say RED, ProRes or DNxHD.  MP4 is more processor intensive, due to its highly compressed Long-GOP nature.  Going from one highly compressed format to another is going to tax your CPU more than your drive chain.

A way to test this, and know for sure if the drive is the issue would be to remove it from the scenario, and see if your export goes faster to a different drive, especially if it's one that's known not to have bandwidth issues.