It was a night and day difference for me. I replaced my Playstation's internal drive with an SSD. Travel times to destinations take roughly 4 to 5 times less than they did on the spinning drive.
While I couldn't replace the internal drive on the XBox in any officially supported manner, I did install an external SSD on it and put D2 there. Saw a similar boost in performance, though slightly less than the PS4 since a spinning hard drive is still running on the XBox for the system software.
Travel times on my Playstation are now comparable to my PC which is entirely SSD. However, menu load times and smoothness are still definitely better on the PC. But the menu times on the Playstation and XBox were still dramatically improved going with SSD.