I'll offer that it's maybe a bit of a trap to argue that one is more important than the other - cue the chicken-or-the-egg paradox...
Weaknesses need to be worked on to the point that they are not liabilities , but strong points in our games need to be polished. I'm probably agreeing 100% with Ash ^^^, but just rehashing his idea with my own lingo.
Andy Roddick certainly developed his strengths to a sky-high level and they propelled him to much success through his career. Roger Federer offers some interesting contrast I think, because when he really emerged and took over for a while, some described his essential strength as actually having no weaknesses. Fed is a rare case, but offers an interesting comparison here.
So Andy pounded the rock against the Fed man, but couldn't really get the better of him... until he worked on some of his relative weaknesses, including his net game and ability to transition forward with authority. Then these two played that epic Wimbledon final ('09 I think?) that proved to be a pure clash of wills instead of one player trying to do the better job of covering his weak spots. I still believe that while Federer won that match, Roddick didn't lose it - that's a whole other rant though.
With that comparison in mind, it's impossible for me to say that improving strengths or fixing weaknesses is generally the higher priority. Every player is a slightly different case, but the Roddick model shows the importance of both. His massive serve and forehand blew many an opponent away and earned Andy lots of hardware. That career of his would have held more disappointment though, if he hadn't gone back to work in the middle of his career to further develop his entire skill set.