Photoshop is a color managed system, while many applications are not. Color management gets complicated real quick. Here is one web site and its attack on the problem. http://www.bythom.com/qadcolor.htm
But basically if you compare photoshop image to the same image with a program that is not color managed it will look different. I don't think Word is color managed and probably not the website either.
But it depends on the color profile of the image and the monitor, so like I said before it gets complicated and above my paygrade.
Curt Y has the root of your problem correctly. To accomplish what you want without opening the color management can of worms, try this:
Open the EPS file in Photoshop. In the Rasterize EPS Format dialog, set the size and resolution you want, then set the Mode to RGB Color. This won't change the way the colors look, but if the EPS is a CMYK file, (and I suspect it is, which would explain why it looks unsaturated on your screen), it will save a step.
Now, open the JPEG version of the logo that's on the web site, (save it from the site if you don't already have a local copy).
With both images open, target the EPS and choose Image > Adjustments > Replace Color.
In a straightforward way, the Replace Color dialog will allow you to sample colors in the EPS and replace them with their more saturated couterparts from the JPEG. If you need instructions, see here.
Save As... (don't overwrite the EPS).