Web images are viewed in the sRGB color space. If you process an image in ProPhoto RGB or Adobe RGB 1998 and then view it through a standard browser (no color management) the colors will shift and generally not for the better. Best to convert all images intended for viewing on the web or in an email application to sRGB during export.
As to why it looks different on the same monitor, LR and PS are color managed applications and take into account the chosen color space when rendering the image. Web browsers and many consumer photo apps on computers (e.g. Photos in Windows) are not color managed and make an implicit decision to treat everything as some form of sRGB. So when they take those colors mapped to a wide gamut space like ProPhoto RGB and display it as though it was actually an sRGB image things tend to get muddy and lose saturation.
That said, when other's view your images on the web or in their email program the colors can again shift even if the image carries an sRGB profile as folks have very different monitors and the settings can vary quite a bit. Nothing you can really do about that part but for your own print workflow you can at least calibrate your monitor, be careful with your color space choices and make sure you embed the appropriate profiles if you send the file out for online printing so they're rendered from the appropriate color space.
In terms of edits, it's hard to say without seeing the image in the appropriate color space but I'd suggest some shadow recovery work to bring out features on the bird and perhaps some effort to bring the sky to a bit more blue rather than gray although that could be a result of the color space issue. It's a shame the lighting didn't give you any catchlight in the eye.
Also be careful of how big you decide to print this as that's a fairly substantial crop on the original image so you've cut away a lot of your pixels.
Nice moment you captured there and the scene with the bird singing and the pine cone is really nice.
