Can a 15 amp and 20 amp share a ground?
It doesn't sound like the #14 and #12 grounding wires being connected is the issue, to me.

EDIT:


  NEC 250.148 (C) Metal Boxes. A connection shall be made between the
  one or more equipment grounding conductors and a metal box by means of
  a grounding screw that shall be used for no other purpose, equipment
  listed for grounding, or a listed grounding device.


The OP uploaded a picture showing that the #12 is an independent circuit, in new 12/2 w/ground NM cable. So what you actually have is grounding via the BX armor, presuming that the BX tightly clamped in both this box and the service panel, and grounding through the #12 wire in the 12/2 circuit, and the two should be bonded together to the metal box.

The ground path through the BX is likely to have higher resistance than the ground path through that nice clean #12 copper. The metal box has to be grounded no matter what. I say leave all the grounds bonded together and to the box, and make sure the BX clamp is clean and tight at the junction box and back at the panel. If you're really concerned about the grounding through the BX sheath, try to run another lone ground wire back to the panel, or pull new NM cable to replace the BX.

Also, you would get a much tighter connection between the armored cable and the box using a better clamp than the one built-in to that old box, something like one of the following (that red plastic bushing protects the wires from sharp edges, so you never energize your grounding system accidentally):






END EDIT

All of the grounds in your house are ultimately bonded together, anyway, to zero the voltage differential all over your house (Neutrals are different! The neutrals are only bonded to the grounding bus in the main panel, nowhere else, and neutrals can only be shared by two circuits under specific conditions).

You wouldn't want a large circuit using a too-skinny conductor for grounding, though.

It sounds to me as if the original circuit is possibly BX armored cable, and the flexible conduit itself is the only grounding conductor going back from this junction box to the panel.

This not up to current code if the flexible metal conduit is the only ground path--there can be continuity problems through the flexible armor. BX was actually demarketed for a while, then came back as armored cable with a metal bonding strip that makes contact with the flexible conduit all along it's length to ensure a good ground path.

However; if you can somehow run a #12 bare or green-insulated wire back from that junction box to the panel (route it up through the attic?) and bond it to the grounding bus inside the panel, as well as to both the #12 and #14 wires, and the BX armor in the junction box by using the right metal clamps and bonding the wire to the metal box, you'd be perfectly fine.

Having said all that, I guess somebody could argue that some grounding through the BX armor is better than no grounding. But if you're in there messing around, you want to do it right and bring it up to code.