The way you explain your issue, it seems like you have an embedded color profile in your image and your preferences are set to keep these profiles.
When you do a "save for web", it usually removes that profile so the image is lighter. And if you do a normal "save as" and look at the JPG in a browser or a software that doesn't do color management, it will look lighter too because it ignores that profile. But when you open that image in Adobe Photoshop, there's a "tag" attached to the image and it tells Photoshop to adjust the colors of that picture with specific values. That's probably why you see the image differently. You need to remove that profile if you don't need it.
I recommend you learn more about color profiles, this answer is just a
very brief explanation of the issue and there's other things you need
to consider when ignoring, using or embedding color profiles!
Example with a profile applied to an image:
The one on the right has a profile embedded, the one on the left doesn't. I used a very dark profile to show the difference.
You need to go in your color management settings
In Photoshop, go in the menu "edit" and select "color settings."
You have some options there, and you should read about color profiles to see what's better for you.
You can set the color profile management to OFF in the "color management policies" section.
You can also checked the boxes that will show a warning when this happens and let you choose if you want to convert the color profile or discard it.
If there's a color profile when you open an image, it will show you a warning.
You can also uncheck the "Embed color profile" when you do a "save as" of your JPG, and re-open it.
Extra info about this:
What should I do regarding color profile when designing for the web?