Firstly, they're both fedoras. I'd always thought that a fedora was a wide-brimmed felt hat with a pinched front, as expertly modelled by Paul Simenon of The Clash
Homburgs are the more formal solid hats with a fixed brim and a centre dent, occasionally called a 'Godfather' hat and here's why....
Lastly, it's not just men that aren't wearing enough hats, don't forget that a masculine hat can look stunning on a woman too. Here's Faye Dunaway (who defined wearing a beret in Bonnie & Clyde) in funky, floppy fedora.
Heres a great site dedicated to hats for any wanting to delve deeper....Headstart