LR and PS do use the same monitor profile, but there have been many cases on these forums where the profile has created problems in one application and not in the other. So try setting the monitor profile to sRGB (Adobe RGB if you have a wide gamut monitor).
If this fixes the issue, you should ideally calibrate the monitor with a hardware calibrator.
If you're on Windows, follow the procedure below.
Press the Windows key+R, type colorcpl in the box and hit Enter.
Add the sRGB profile, then set it as Default profile. See screenshot (from Windows 7) below.
Make sure that Use my settings for this device is checked.
I am using Win 7 and I have done as you instructed but the issue remains. I've also installed my monitors' drivers (two screens - they were plugnplay before) and of course restarted the machine several times. Are you aware of anything else I could try?
Actually, installing monitor drivers is usually a bad idea. No actual driver is necessary (plug'n'play works), and about the only difference is that the model name pops up here and there.
The real problem is that it also installs monitor profiles, and manufacturer monitor profiles are frequently so bad that you are much better off without them. Indeed, manufacturer profiles are usually the direct cause of these problems.
I still think, as Per does, that this is a bad monitor profile. So until you get a calibrator to make a proper profile, go back into Windows color management, throw out any manufacturer profile, and replace it with sRGB. Or Adobe RGB if this is a wide gamut unit.
Relaunch Photoshop/Lightroom when done. They need to load the new profile at startup.
Any difference between PS and Lr is almost always a monitor profile problem. Any two color managed applications should always display identically, whatever the color profiles used. That's the whole point of color management.
One reason the two apps may react differently to a bad profile is that the source profiles are different - Lightroom uses gamma 1.0 ProPhoto, while in Photoshop the file may be in sRGB, Adobe RGB or gamma 1.8 ProPhoto. So even if it ends up in the same monitor profile, the two actual conversions are different.
This colour inconsistency has cropped up suddenly and I don't understand why. I even re-calibrated the monitor and made a new profile, but no change. The colour space in both apps is ProPhoto. I'm at a total loss here!
I may be wrong, but it seems like the issue crops up with certain images only, not sure why.
Lightroom Desktop CC
I actually believe this was due to a bug. I was actually having the same issue. It's bizarre, as if I exported to tiff or PSD to edit in PS using profoto RGB profile, it massively changed the colours. The image looked so different. I set the colour space in PS CC to be profoto rgb too. The way I worked around it was to export to a jpg then edit in PS (simple image cutout was all I needed to do). It seemed to handle the 8 bit sRGB image fine.
Anyway, I just updated PS and the issue has now gone!