When I let Photoshop control color, the print colors are virtually identical to what I see on the display.
Among the various profiles available in Affinity are:
- RGB
- Apple
- Adobe
- Colormatch
- Display
- Display P3
- Generic
- ROMM
- SMTP
- Thunderbolt Display 2
- sRGB IEC61966-2.1
- 32 Bit RGB
- ACES
- Adobe
- Apple
- Colormatch
- Display (linear)
- Display P3
- ...
- Thunderbolt Display
- CMYK
- Euroscale
- Generic
- several Japanese variants
- U.S. sheetfeed coated
- U.S. Sheetfeed Uncoated
- U.S. Web coated
- U.S. Web uncoated
Then there is Rendering Intent:
- Perceptual
- relative colorimetric
- saturation
- absolute
I suspect that by proper selection of profiles you can get as accurate color rendition as Photoshop, perhaps by using Adobe's own profiles. As I said, I have not really tested printed output as most of my work is for the web.