Though there are a few similarities, these are different windows for different uses.
Another difference is the way they open and close.
Casement windows are operated by a manual crank on the house interior.
The Spruce / Brianna Gilmartin
Turning the crank engages a gearbox that pushes the window open or shuts it.
Double-hung windows have both an upper and a lowersash(windowpane unit) that slide up and down.
The user holds a handle and lifts either sash straight up or down.
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Normal window unit air conditionersdo not fit in casement(and slider) windows.
More expensive specialty A/C units are required.
Gusts of wind can catch a casement sash and rip it away.
Poorly designed casement window placement can result in conflicting windows.
For example, two casement windows' sashes on an inside corner may collide with each other.
Double-hung windows are reliable and have a low failure rate.
As long as you have close access, the double-hung window is fairly easy to operate.
A spring-loaded balance aids in lifting the window and gravity helps you close it.
But if you have to stretch to reach the window, it can be exceedingly difficult to operate.