What size is the PSD (in pixels)?
What resolution (PPI) does Illustrator say it is in the links panel?
Are you viewing it at the same zoom level in Illustrator and Photoshop?
At which zoom level are you zoomed in?
Did you scale the image in Illustrator?
Please show screenshots of the pixelization.
Do you check the image quality in AI or Acrobat?
AI shows the rasters with a decreased resolution to minimize the system charging. Is the size of your PSD the same in AI as in Photoshop? Is the image resolution 300PPI?
Yes, it will be printed better than it looks in AI. The appearance of different kinds of imported files may differ depending on their origin (PDF and PSD do not look the same). It doesn't matter.
If you want to print your artwork in the general printing house, I would suggest to change the image size in PSD so that the final resolution (the one that you can see in the Links panel in AI) become 300PPI. It provides the best printing quality.