When purchasing from the Mac AppStore a new 'unlocked' copy of the app will be downloaded. If there is already a non-AppStore version of iThoughtsX installed (a trial perhaps?) then the purchased copy will be installed alongside it in a folder called iThoughtsX. It is important to delete the non-AppStore version once the AppStore version has been purchased. Indeed, if you launch the non-AppStore version it will probably complain about 2 copies being installed (as below:)
If you choose 'Quit and Show in Finder' it will take you to where the non-AppStore version is installed - enabling you to quickly delete it. Another way to locate the non-AppStore version is to right click on its icon in the Dock and choose 'Options -> Show in Finder'. See below for a typical setup.
Above you can see the non-AppStore version and a folder of the same name containing the AppStore version. To uninstall the non-AppStore version, just drag it to the Trash. Remember to right click on the Trash and choose 'Empty Trash' after doing this.
With the non-AppStore version gone, you will need to go into the iThoughtsX folder to find the iThoughtsX app and launch it. To keep it 'handy' in the Dock, launch the app then right click on the icon in the Dock and choose 'Options -> Keep in Dock'. Alternatively just hit 'LaunchPad'
NB: There is no jeopardy here - you can alway redownload either app - and (unlike iOS) deleting an app doesn't delete your maps - they're safe.