Congratulations, you've created an open (floating) neutral.
"I omitted the neutral and ground from the main panel... I hammered a new ground rod and bonded the ground and neutral at the new building sub panel, which I figured would give me the same setup as a main panel..."
This is partially incorrect. The main panel should be supplied from the utility with two ungrounded (hot) conductors, and one grounded (neutral) conductor. You'll then have an on-site grounding electrode (ground rod, etc.), which the equipment grounding conductor and grounded (neutral) conductors will be bonded to.
To feed your subpanel you'll have to provide at least two ungrounded (hot) conductors, and a grounded (neutral) conductor from the main panel. Since this is a separate building, you are correct that a separate grounding electrode is required. However, you'll want to make sure it is an adequate ground before using it. It's not likely you'll have the equipment to do this, so you'll probably have to call somebody in to verify a good ground.
If there are no other bonded metallic paths between the buildings (water pipe, gas pipe, coaxial cable, etc.) then you're allowed to simply run two ungrounded (hot) conductors and one grounded (neutral) conductor. In this situation, the grounded (neutral) conductor should be bonded to the grounding electrode at the subpanel.
If there are other bonded metallic paths between the buildings then you should run two ungrounded (hot), one grounded (neutral), and one grounding conductor. In this case, the grounded (neutral) conductor will not be bonded to the grounding electrode at the subpanel.
See: National Electrical Code 2011 250.32(B)(1)
Ground and neutral are not parallel neutrals. I know it looks that way because they're bonded in the main panel. But shift into a different way of thinking about the purposes of the 2 wires. Think of the ground solely as a safety shield.
Let's try a few pairs of examples. The first is Code and the second bonds at the sub-panel also. The orange glow is on things which are "hot".
Seems awesome right? Poor old Code Man is in the dark. His power tried to return via neutral, and neutral is broke, so the power failed. Rogue Man is one happy guy and his life isn't disrupted. Ground is working great as a "backup neutral". He doesn't even know he has a problem!
Of course, ground is a thinner wire, so it might overheat, but so what? Or, what if both ground and neutral were cut?
Code Man is still in the dark and he's still gotta fix those wires. Rogue Man is dead.
In Code Man's installation, the hot went through the bulb, looking for neutral. It didn't find it, so it pulled the neutral up to 120V ( not enough power for useful work, but plenty to shock). It did the same for Rogue Man, but since he tied neutral to ground in the sub-panel, ground is now also 120V, including the service panel cover and the switch plate cover screws.
Suppose the sub-panel has its own ground rod. That doesn't help much. Earth tends to have high resistance, so the cover screws might be 103V instead of 120V.
I have the good fortune of working in EMT conduit in a steel building, which naturally forces the entire conduit system to ground. Ground is never part of the circuit in any way whatsoever. So I get to see it as intended, as a protective "shroud" around all things electrical.
Ground isn't quite yet a perfect envelope. It is in new work, but we still have a lot of old wiring out there that is not practical to outlaw entirely - such as NEMA 10 and switch-loop smart switches which poach ground as a neutral.
Why bond neutral at all?
That's a GREAT question. Not bonding ground would give you an isolated system. And that makes a lot of sense in some ways, like solving some of the problems you see above. But it has other disadvantages. I go into depth about that here.
Best Answer
Short answer, you do want a ground conductor between the main panel and subpanel (not just a ground rod at the subpanel) and you only want the neutral bonded to the ground at the main panel.
The ground rod alone is not sufficient - although there would be a path through ground back to the main building's ground, that is not a low impedance path, it wouldn't be low enough to clear a fault, not safe, etc.
The ground conductor could be a metal conduit from panel to panel provided it's installed properly so that it creates a continuous path back to the main panel.