I would like you both to try some things:
Right-click on the 0.4.0 application, choose "show package content", open Contents/MacOS and click on PyxelEdit. Did anything happpen?
At the same time open Console.app, and use the top right bar displaying "search" type Pyxeledit and insert the log here.
Since you have the older version working, you should have a PyxelEdit or PyxelEditPR folder located in the Library/Application Support folder now. However, if you're unfamiliar with the OSX environment, the Library folder will be hidden, you therefore have to either use terminal.app, and write "Open /Users/[username]/Library/Application Support/PyxelEdit/" or open Finder app, go to the top bar, click on the "Go" tab, hold down alt/super and the library folder will reveal itself.
If you do have a "PyxelEdit" folder now, try to delete it and open the application again. You'll have to insert your key again.