I'm having a hard time picking between those two cases. And I don't know if the H440 is still a good choise when it came out kinda long ago than the S340 did. I will be using the Corsair H100i V2.
IfI had to choose then the S340 would be my choice. It depends on how much space you want inside and how much space you want the case to occupy on/under your desk. The tempered glass on the S340 Elite is a winner for me.