In terms of your different perception of brightness of your eyes, it could be the case that you have a fixation issue so that you doesn't look with the part of the retina which should be used so different light receptors are used.
I think as Mike outlined before, that many of us have in general trouble with our binocular vision which is in different forms triggered by different displays. If you have trouble focusing you probably more inclined to use parts of your peripheral vision to compensate.
However bright and sharp displays with strong contrast ratios make it more difficult to do so, since they stimulate fusion which is more taxing on your eye muscles if you have trouble with your binocular vision. So it is not really the displays fault here but an unfortunate consequence of better displays, antialiasing or other software improvements.
If you have some success with one eye in the beginning, it is not unlikely that you are nearly symptom free in 3-6 months during the time when you have one eye covered. As Mike said for most people it is better to cover their non dominant eye, but just try it for yourself. You should feel a preference after some testing. I would also suggest to only cover your eye in situations in which is necessary to do so to keep you binocular vision alive, but most people do that anyways.
However, in most cases covering one eye should not help you with your binocular vision, but it can.