They're both really good in their own way. Trello is good for having an overview of what's going on, and the design is almost tactile. It's a very satisfying feeling, dragging a card to a "Done" list.
But Asana has its own strengths. It's better in some ways for actual task management; not the bigger chunks implied by Trello's cards, but the nuts and bolts, the individual actions it takes to get something done. It's really nice to be able to start the day by going through projects and marking a few tasks as "Today". And assigning arguably works better - the way that an assigned task shows up in your inbox (alongside your private tasks if you want), and you can choose when you're going to do it.
There are workarounds in each - you could set up a "Today" list in Trello, or use Asana's tags for more advanced workflow.
Probably the best thing to do is try each and see which one feels better for the stuff you're doing.