I took a listen to the song and I can say that there is not a lot wrong with the recording......especially as it is an amateur home recording. You seem to have the basic skills down pat and what you produce is pretty good as a good demo or worktape. OK you may have a bit to go to produce radio ready fully mastered tracks BUT most people cannot achieve that without thousands of bucks worth of equipment and the tech skills to match.....to achieve that is a very long curve of learning indeed and a lot of people would use the experts like a pro studio or specialist for mastering etc. I would agree with Colin that most DAWs are similar and will all produce good results when used properly BUT I doubt very much if Logic Pro or even Reaper will make much of a difference quality wise to your results. What you may benefit from is some more tech expertise on how to get the best from what you already have. This may involve some sound engineer classes or tutorials and reading specialist books or online courses. There may be a local college or studio that can offer these....or Google.
Remember a good sound engineer can get great results from pretty average equipment but someone who does not have these skills has no chance regardless of the quality of gizmos at his fingertips. In short to reach the next level it is your skills that need an upgrade not your equipment. That will take time and is a big learning curve.....but hey you are well on your way already.

http://www.soundclick.com/bands/songInfo.cfm?bandID=687727&songID=5238055