From my point of view, I see Trello better suited for managing a team project, while I see Asana better suited for handling personal tasks and projects. Trello allows you to assign multiple members to one card (task, depends on how you used it), while in Asana you can assign only one person to a task and followers get an update. Trello does a great job of providing the big picture of a project and its progress through its unique UI. Though the UI is not all rainbows and cookies. The larger your monitor the better because you have to keep scrolling to see cards.
Right now I use Asana to manage everything personally, but I use Trello to manage our company. If it was up to me I would like to do everything in Asana. I would like to see Asana give mobile a good shot. Right now they have a 2-star iPhone app and no Android app. Trello on the other hand as both iPhone and Android apps, and they are work pretty well.
One feature I really like about Asana is that it can sync your tasks to your favorite calendar app without any fuss. This feature makes Asana perfect for managing personal tasks with ease. Both services are mature but still have some way to go.