Well, at least right now, 1GB+ cards are the high-end ones which require more than the 250MB/s transfer rate the 1x slots provide. The number (1x, 2x, 4x, 8x, 16x) refer to the number of lanes going said PCIe slot. This is independent of the 1.0, 1.1 or 2.0 PCIe revision. It's been awhile since I read up on it, but 1.0 and 2.0 can have some compatibility issues at the hardware level. However, 1.0 hardware is pretty old and not as common. Besides a higher transfer rate 2.0 offer, they also allow for more power to be delivered via the PCIe connection.
As far as connectors for 2.0, the only thing I can think of is the lack of an 8-pin PCIe power connector, which is present on high-end cards.
Of course, if this is a server, they're typically not designed with a high-end graphics solution in mind, so finding the accommodations are generally lacking.