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).
adb push will copy any file to the phone, whereas
adb install will only accept an apk file, and will install it onto the device.