Agree with Golf, this may be hardware like the power switch or perhaps usb port. (but it should still turn on if it were usb!!!)
Only thing I could recommend:
0. check the contacts between the phone and battery
1. check your charging cable and your (wall) charger as well.
2. plug the working phone into the charger cable/charger and see if it switches to charge.
3. Replace cable and or charger if they don't appear to be connecting or charging correctly,
4. If they do seem to be working well, try turning the phone on while connected to the charger
5. If that does not work, make sure the dead phone has a reasonably charged battery (> 30%) (yours!) and then try starting it /placing it into the recovery mode.
-do a reset by Pressing and holding the Volume up + Power + Home buttons simultaneously. See if you can get the Samsung logo to appear.
- If you can, release the buttons and you should be in the stock recovery. (You can reset the phone or just restart at this point).
6. If the cable check and trying to get back to recovery both don't work, then I would recommend returning it to store for servicing or replacement as well.
Good Luck!