Hello,
I have had both, well sort of. The HyCoSy I had "failed" as in, they couldnt insert the tube into my uterus to get the dye in. It turns out I had scar tissue blocking the cervix, I later had an operation to remove this, and then had an HSG to check all was OK.
The big difference between the two is that with the HyCoSy, they put saline solution or contrast dye into your uterus and then do an ultrasound and with the HSG they put contrast dye in and do X-rays.
This usually means (but not always, do ask) that HyCoSy's are done in a doctors surgery by a gynecologist or ultrasound technician, and an HSG is done in a hospital where there is large xray machines (and will involve lots of people, nurses, technicians, gynecologist etc)
Both exams IMO are quite painful (try to take the strongest painkillers you can get your hands on before hand) and humiliating - you spend a lot of time in stirrups while people poke around. In the US it is quite common for gynecologists to offer women some kind of tranquiliser for these procedures as well as painkillers, but not so on the NHS!
If you know your cycles well, try to get an appointment for mid cycle when you are likely to be ovoluating and your cervix will be a little open anyway.
Hope that helps. HTH xx