Why do I see two icons for the same application?
There are two things this could be.


You may be running two different instances of TinyGrab. Spotlight ignores quite a few locations on your Mac. To see where they live, right click each one and choose "Show in Finder" (it may be under options depending on your Mac OS version). If there is a second application, there is your problem! ;)
You are in fact running two instances of TinyGrab. The default is to only run one instance of any GUI application, but there are options to disable that. To see if this is the case, when both are running open Activity Monitor and search for the program. If you see two entries there, that is the issue. If this is the case you can check the preferences to see if there is a multiple instance option (I didn't find one).


Finally you can also try deleting ~/Library/Preferences/co.uk.keyoneproductions.TinyGrab.plist and see if that solves the problem. And if that doesn't, try removing your Dock preferences at ~/Library/Preferences/com.apple.dock.plist. Maybe try deleting ~/Library/Preferences/com.apple.loginitems.plist as well, then logging out and back in?