You've most likely stripped the profile, using Save For Web or Export. For some unfathomable reason this is still the default setting in both modules. You need to manually check "embed profile". A regular "save as" preserves the profile.
When you bring an untagged file (without profile) into Photoshop, the working space kicks in whatever it happens to be. If the file was created in a different color space than the working space, anything can happen.
As long as the profile is embedded, that will always override the working space (unless you've changed any color settings, which you should never do until you know what you're doing!).
Adobe's policies on this are strangely schizophrenic. The graphic applications have always been the industry standard in color management - but as soon as it comes to anything web-related, it's as if they never even heard about color management. Haven't they registered that most major web browsers have been color managed for ten years by now? It makes no sense.