This looks like a color management issue. Photoshop is color managed, meaning that it looks for the ICC profile attached to the image and uses that to help ensure an accurate rendering on your monitor or viewing device. Fireworks is not color managed, so it ignores the ICC profile. This is analogous to how the image might appear when using "Save For Web" in Photoshop and choosing the Uncompensated Color view option.
Some browsers like Safari and Firefox are now color managed, so if an image has an attached profile, it will be rendered more accurately, much like in Photoshop. This wasn't always the case with web browsers, which is likely why Fireworks was built this way.
I'm using an older version (FW8); I'm not sure whether they've added an ICC profile option to Fireworks's export/optimize panels in newer versions. Also I don't know whether an ICC profile is actually *removed* from an image when it is saved in FW, or if the profile is simply ignored—but you could test for this. (Note that in Photoshop's Save for Web, an ICC profile option is included for images being saved in JPEG format.)
I should have added that you can check the existing ICC profile of your stock photo in Photoshop, using the Info panel. If the profile is anything other than sRGB, consider using the Edit > Convert to Profile... command and converting it. My current RGB working space is set to "sRGB IEC61966-2.1", which is a good profile for web use—unless there's a more recent profile that's being recommended for this purpose.
Sometimes, I use Mac's Preview application to make quick color profile adjustments. As an experiment, I took your first image and assigned it a "ProPhoto RGB" color profile—this is a wide gamut color space. In a color managed environment, the image now appears overly saturated and pink-ish (pretty gross!). But in Fireworks, it looks exactly like your original. As a test, I then saved this image as a Fireworks PNG; the profile was omitted by Fireworks in the newly saved version and so its appearance was just like your original.
Here are the two images: 1) ProPhoto RGB and 2) Fireworks PNG (without ICC profile).
The only difference between these two is the ICC color profile, or lack thereof. If you don't see a difference, then your browser isn't color managed. (P.S. I just discovered that in this forum, you need to click on the image—which opens up a lightbox-style viewer—to see it with color management applied. That is, if your browser supports it.)