There are three classes here:
1) good PCI cards, can be on a PCI card or onboard.
2) bad PCI cards, can be on a PCI card or onboard. Same as 1) but higher interrupt load, e.g. most (all?) Realtek chips.
3) non-PCI connections.
4) higher speed PCI cards, PCI-X, PCI-E.
The important part here is to understand that most, almost all, onboard ethernet cards on mainboard are actually connected via the PCI bus. Although they are not on a PCI card, they are routed through the PCI bus on the mainboard.
Only solutions like Intel CSA and NVidia's integrated Gigabit in the nForce3 250Gb are not on the PCI bus.
However, there is not strictly a faster versus slower. A normal PCI bus has enough bandwidth (132 MB/sec) to support Gigabit ethernet at full speed (108 MB/sec). However, the PCI bus is pretty much full in that case. If you also want to use harddrives at the same time, then both harddrive controller and ethernet share the same bus and then things get "slow".
So, only if you use gigabit ethernet at the same time as the harddrives you need non-PCI ethernet. Typically, this is the case for network servers, which get data off the disks and deliver it over Ethernet.
If you just consume the data that comes in over the gigabit connection in the CPU and don't use disk or other high-bandwidth I/O at the same time, then you will not get a big advantage over non-PCI Ethernet.
Hope that helps, let me know if something is unclear.