Printing on an Epson R220 and using the downloaded Ilford ICC. I've chosen ICM and No Color Adjustment in the Epson Properties box. I've got some gaps in my understanding - Why use sRGB (which appears to be default setting for cameras) when printing uses aRGB? I should just go back to film.
It's all RGB color space just different versions if you will. Some RGB spaces favor blues some favor reds some have bigger gamuts than the printer can print (like ProphotoRGB). I don't know the difference between aRGB (Adobe RGB?) and sRGB. If aRGB is Adobe RGB both these spaces are similar enough that you should get the same overall color.
Go to Joseph Holmes website. He has all the info on RGB color space.
Go to FAQ's and look at "How do you use RGB spaces..." talks about RAW conversion etc.
The biggest differences when printing are what driver you use (Epson, or custom RIP) and ICC profiles for the paper. Sounds like you have the right ICC profile.
Hope this helps, this is a much bigger topic than I can talk about. Maybe some of the more technical folks will respond as well.