images look desaturated

What are the Edit > Color Settings in both applications? 

Please set the Status Bar to »Document Profile« and post meaningful screenshots with the pertinent Panels (Toolbar, Layers, Options Bar, …) visible.

@Tatyana34396649uz7f  IF tyou have embedded ICC profiles as D.Fosse mentioned then this could be a faulty monitor profile, such an issue might affect different applications differently. 

Have a look here where that was discussed and a solution provided:

https://community.adobe.com/t5/photoshop-ecosystem-discussions/photoshop-and-illustrator-colours-are-completely-different/m-p/14457611#M782757

I hope this helps
neil barstow, colourmanagement net - adobe forum volunteer - co-author: 'getting colour right'
google me "neil barstow colourmanagement" for lots of free articles on colour management
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Make sure the color profile is embedded from Illustrator. The embedded profile should always travel with the file, and it will (or should if it works correctly) override the color settings.

AI induces some color noise in the image in addition to washing out some of the color. Have you tried one of the Blend Modes in PS? You can also check the Image Mode.