The colour mode of your image file is CMYK. The colour profile is U.S. Web Coated (SWOP) v2.
Since the colour mode of displays is RGB, a browser cannot directly show this image. A browser that does not support colour management (like Internet Explorer 8) may not be able to show this image at all.
However, various browsers support ICC profiles (ICC version 2 or even version 4). A compatible browser can use the embedded colour profile (U.S. Web Coated (SWOP) v2) and convert the data to the RGB colour mode and to a suitable colour space. The destination space usually corresponds to the sRGB profile or preferably (on a calibrated system) to the colour profile of the display.
In this case, the conversion may yield mixed results. For example, the image looks good in Internet Explorer 11 and Safari but weird in Firefox. The precise reasons are not clear since the conversion is an internal process of the particular browser. The various conversion options may be different from browser to browser. Furthermore, in this case, the browser has to convert not only the colour profile but also the colour mode.
There is another reason why your image is a special case: Some shades of cyan are outside of the sRGB gamut. Thus, the browser cannot correctly convert the given CMYK values of these colours to corresponding sRGB values. Therefore, the choice of particular conversion options may be especially important for these colours.
In order to solve these problems, you may want to convert the image to the RGB colour mode and to a suitable colour profile yourself (in Photoshop), so that you can control the various conversion options, and so the browsers do not need to do conversions with uncertain results later.
For web presentation, you should usually convert images to the sRGB profile. In this case, though, the sRGB gamut may not be suitable to cover the shades of cyan in your image. Therefore, you may want to consider using the Adobe RGB profile, which covers these colours. However, browsers that do not support colour management (like Internet Explorer 8) cannot correctly display the colours of images that are using the Adobe RGB profile.
In any case, make sure to embed the used colour profile. Even if you use the sRGB profile, the browser may still need to convert the image to the colour profile of the display.