Actually not just the soft proof but it's a bigger difference there.
So, I open the same image (a tiff) in both Lightroom and Photoshop (working space Prophoto) and on the same calibrated screen(obviously). The image in Lightroom is slightly (but noticeably) more saturated.
Then with soft proofing implemented in both, with the same profile (in this case Ilford Galerie Smoot Pearl), same relative colourimetric intent and simulate paper and ink implemented in both. There is a much bigger shift in Photoshop than in Lightroom (Photoshop reducing contrast and saturation and slight hue shift, Lightroom very slightly reducing contrast and very slight hue shift). Photoshop sofproof is a much closer match to the print.
Why the difference?
Is the initial slight difference due to the different working space? But more importantly why the bigger soft proofing difference?