Android - What's the difference between 'adb push' and 'adb install'?

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.

More specifically

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).