Did you measure (in Photoshop) what you call ‘pure black’?
Is the Photoshop file in CMYK or RGB?
Is the black in Illustrator built up in the same way as the ‘pure black’?
Is the Illustrator file in CMYK or RGB?
Is the photo linked or embedded?
Are you using Overprint Preview?
Take a careful look at the colours in both your files and I think you will be able to find the answer yourself.
Synchronizung colour settings in all Adobe programmes simultaneously is best done in Bridge.
Cmd+Shift+K on a Mac, Ctrl+Shift+K on Windows.
The colour profile you choose for converting RGB to CMYK (for printing) depends on several factors, printing method and paper quality being the most important ones. So read up about colour management and/or contact your printer for specific details.