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)
Since there is still some uncertainty here, I took a minute to create an image that might fix some of this.

Lightning is a DC shot of electricity going to the ground, it's source. So it's easiest to consider it that way. Next, lightning is so powerful that it creates a surrounding zone of energy on any conductive material that is able, the yellow circles. Everything that is within a distance becomes energized to a point, but is again still from the lightning and is trying to go one way - to the ground. This is why everything metal is bonded or grounded: so that nothing needs to arc to find a path, since it has it's own. Regardless of how it gets to ground, it's going there.
Back to the electrodes though. Consider my image above as being of a main structure and a garage. For all purposes necessary with lightning, there is no need to tie them together. As mentioned above about 100,000A of power coming off of a lightning strike, more rods and such can help dissipate the power faster and catch a little more of it than is going through metal in your home, but regardless it's still 100,000A of power ... what happens, happens.
Whether it's your garage or your neighbors house that is splitting the distance of the strike zone, tying them together won't fully relieve the problem of getting struck by lightning.
For purposes of the code, which you fully understand, the connection that is made between the two of them is your EGC that is only for ground faults. The reason this is in the code is because some people think that running a ground rod will be the miracle solution to ground faults and that the wire isn't necessary. This isn't the case though.
For ground faults, the EGC is the answer because it sends the fault back to it's own source - the utility. For small voltage differences throughout a system, a ground rod (or multiple for higher systems) is driven. However, for lightning, you are again giving the ground rod as a path for the voltage difference caused by it, but tying the full system together doesn't help all that much more than one rod.
Hopefully this answer helps a little better.
Best Answer
You only have to bond the closest one to the panel. NEC 250.52.a.3.2. (2017) same in the 2014 code.