In addition to CMYK being somewhat obscure and in addition to possibly applying "some" color profile, the web preview might also be different because of scaling. Scaling is not necessarily gamma-correct.

CMYK by itself is a bit of a weird, somewhat unscientific beast, too. Actually, it's CMY, not CMYK (adding up absorption instead of adding up light -- subtractive color system). However, although full-intensity on all three components in CMY should give black, this is rather expensive (uses a lot of ink) and due to technical limitations in practice it doesn't look quite right. Meaning that it's kind of black, but not really black, and you don't get 100% sharp black contours due to imperfect overlap.

Hence, what you do is you throw in a fourth channel (key plate, hence "K") which is just pure black, and reduce cyan, magenta, and yellow colors by some amont (found via trial and error) so overall brightness seems about the same, and it "looks alright". No surprise that somehow it just fails to 100% reproduce something from a different, more physically oriented model.

More data.

If I open the downloaded file with chrome or firefox, (3) is shown.

If I use the other applications, I see (4).

Edit: Yeah, those two browsers actually modify the image to what they think it should look like. (4) is the correct cmyk.

To further complicate things, monitors are calibrated differently. What you see and what we see is different. What I see on one monitor/computer and what I see on a different monitor/computer can be dramatically different.

When I make a piece of artwork, if I print it, even if the printer perfectly reproduced the computer's colors, it still might look wrong, because my monitor may not have perfectly reproduced the computer's colors, so I likely even created the digital art using the wrong colors, because I was basically painting through the wrongly-calibrated monitor.

Even worse if I draw a piece of art in real life, scan it (changing colors), view it on my monitor (viewed wrongly), print it (changing colors). It can be darker, lighter, greyer, more colorful, more contrasting, less contrasting, less blue, more green, red can look pink, and yellow can look white, etc...

Digital stuff is fantastic. Ones and zeros are ones and zeroes. But when digital interacts with the real world, the conversions are very approximate unless you get expensive hardware and properly calibrate them. One place where "digital" meets "real world" is when digital values are converted to light for the monitor to display. This doesn't even yet involve RGB -> CYMK conversions, and already things vary wildly.

When making games for consumers, you have to remember that the billions of consumers in the world have tens of thousands of different screens, manufactured by dozens of different companies, set to millions of different settings... And then the ambient lighting of their living-room/whatever affects things, and even if everything was perfectly the same, different peoples' eyes perceive colors differently anyway.

I don't know the solution to this. It seems there are things you can do to help, but I can't conceive of a perfect solution.

The profile is not all that's needed. You generally specify how the conversion should be made: perceptual, absolute colourimetric, relative colourimetric etc.
That website does not allow you to set this, and it can have a negative impact on the image.

If you or someone else is still looking for a way to convert your sRGB image to a CYMK profile and you don't have Photoshop ($$$) or Krita (free, but needs a powerful system), you can use ImageMagick:

Download either of the portable versions (the last two at the end of the list):
http://www.imagemagick.org/script/binary-releases.php#windows

Then download the ICC profile pack from Adobe:
https://www.adobe.com/support/downloads/detail.jsp?ftpID=3680

Then unpack ImageMagick to some folder, open the command prompt and navigate to that folder. Then use this command line:
convert rgb_image.jpg -intent relative -black-point-compensation \
-profile "CoatedFOGRA39.icc" cmyk_image.jpg

With "CoatedFOGRA39.icc" being a file from that profile pack from Adobe that you copied to the same folder as ImageMagick's "convert.exe".
More information here:
http://www.imagemagick.org/Usage/formats/#profiles

EDIT: The choice of a rendering intent (perceptual, relative colourimetric etc.) depends on what you want from the resulting image. From what I understand of it, for colours outside of the CMYK profile gamut, the perceptual mode preserves value nuances but may shift hues, and the relative colourimetric mode preserves hues but may lose value nuances.
- http://www.color-management-guide.com/conversion-mode-perceptual-relative-colorimetric-rendering-intent.html
- http://www.steves-digicams.com/knowledge-center/understanding-rendering-intents.html

PNG (and probably other modern formats) sometimes include a "Gamma" setting in the image that can mess with brightness. The pixel data remains the same, it's just that Gamma setting that causes it to be displayed differently in web browsers.

This is why I always remove the "GAMMA" chunk from my PNG images, I use TweakPNG for that.
http://entropymine.com/jason/tweakpng/