I tried to match up the color management<<
From Photoshop's Edit > Colour Settings, SAVE your current settings to a colour settings file (.CSF) to the default location. From Illustrator's Edit>Colour Settings d/box LOAD the .CSF file you just saved. Now you can be sure both applications are using identical settings.
However Pantone colours and RGB really don't go together - Pantone colours are representations of inks which is very different to the RGB colour space. Photoshop's conversions will be more accurate than Illustrator's since Photoshop's Pantone libraries are reference the L*a*b colour space whereas Illustrator's reference an unknown CMYK colour space.