I do not use Flickr myself and looking at the page it appears they are using some method to stop me easily downloading the image to check. To try and find out what is happening here I just dragged the small image on this page to the desktop and opened it in Photoshop, where the profile is shown as Adobe RGB. Then I did the same with the bigger image on this page and the profile on that is ProPhoto RGB.
Both of these are unsuited to the internet and you would be far better off using sRGB. I would concur that the small file looks a good stop darker than the image on Flickr with subtle differences across Chrome, Safari and Firefox on my Mac.
At time of writing Firefox appears to be the best browser for colour management, but since browsers are in constant development that might not always be the case. It's also true that you cannot govern how others are viewing your images, but sRGB is the safest option across all browsers.