Hextech Gunblade vs Lichbane

Depends how much AP you have. With 1000 ap Lich bane does more damage, but with most normal amounts of AP gunblade does more damage. There is an important difference, though Gunblade active has a 40 second cooldown, while lich bane has only a 1.5 second cooldown so you can get 2 lich bane procs off in 3 seconds (and 3 in 4.5 seconds) while you only get one gunblade proc per fight.

So lich bane is a lot better for sustained damage, but gunblade provides health sustain. Lich bane is better for bursting unless you can use the AD from gunblade well as you can cast a spell, attack with lich bane, then repeat 1.5 seconds later to outdamage gunblade.

Gunblade is mostly for hybrid champs/sustain champs and has a powerful slow, while lich bane is better for AP stackers.

All you need to do is look on the wiki for these things. And why not build both? ;)