While no expert on this subject is there a reason for using CMYK? The posts I read indicate use RGB fpr edits and convert to CYMK on last step if needed. Even then colors may not be exactly the same.
The "K" channel in your Illustrator file is Process Black Ink. As you, see it is lighter than R-0, G-0, B-0 in your Photoshop file. (You can have this "Rich Black" in Illustrator, but it will be a mix of C, M, Y and K inks)
You can recreate in AI in RGB mode, or use Levels in Photoshop to darken down.
But as Curt asks, is there a reason you are creating original in CMYK color mode?
Yeah it's a document for print so didn't want to work in RGB then change the colour mode to CMYK at the end in case it changes the look of the colours
it's a document for print
Print where? Only off-set printing firms use CMYK process inks.
All desktop inkjet printers expect RGB numbers, many have a wider gamut than CMYK. It does no good to work in CMYK if the output is inkjet.
A better method is to work/edit in RGB and Soft Proof to your output profile. Best of both worlds that way.