adb push only copy the file into the target location whereas
adb install copies the file into the target location and also sets the proper permission for you?
adb push [file] [path]
will take [file] from the PC and copy it to [path] on the android device.
adb install [apkfile]
will copy [apkfile] from the pc into
/data/app/ on the android device.
The main difference being that push will let you specify where you want the file to go on the device, and install puts it in a known location (the location "installed" apps exist in).