To be fair about this, I'd like to compare both of them in accordance to how they started out, up to the point in where they currently are.
May started out having no interest in Pokemon, However, when she saw and fell in love with Coordinating, you can see that even after that, she really didn't put much effort into her first contest, probably from underestimating the opposition, to stage fright or simply having little idea of what she should be doing.
Dawn however, started her journey with the firm intent of being a Coordinator. Albeit she wanted this mostly because of her inspiration from her Mom, but in any case, both of them were inspired to take up Coordinating. We can't really see how Dawn practiced for her first Contest, but in any case, she did have the enthusiasm amidst a wee bit of stage fright. She wasn't even afraid of the opposition going even further as to ask Ash to compete as well (Although that could be support or simply Dawn's want to fulfill Aipom's wish of being in a Contest.)
Ever since then, both have gone through a series of parrallelisms, both won their second contests, both experienced a state of arrogance at some point, went through some inner turmoil (although this mostly affected Dawn who suffered two losses in a row) before both coming to terms with themselves and finally competing and winning yet another contest.
From here on out, Dawn has yet to develop, and it is uncertain what will occur during the Celestic Convention. May however, proceeded to win some, and lose some, however she was able to look at her wins and losses with calmness and maturity. (at least as far as May goes.)
Having looked at both cases, I'd have to go with Dawn. I consider her strong in the sense that she was able to bounce back after her losses, but didn't go so far as to underestimate her future contest rivals like May did. It is notable that May has managed to defeat her rivals at some point, while Dawn has yet to defeat Zoey. But in her magnificent battle against Kenny and May, one can sense a stronger will from Dawn than I think May could ever express. It might have something to do with more emotional character development on Dawn's part, but one can really relate to Dawn's drive to win.