Sure, Asana does task management... but that's about all it does. Slack is a lot more casual, and you can create channels for each project and have conversations there about who will be doing which task.
Internally, we actually use Trello for task management. Honestly, it's far superior to Asana in terms of ease of use and flexibility. It has less features, but that just encourages keeping things organized in simple ways, which is great.
You find Trello easier to use and more flexible than Asana? I actually find the exact opposite. You do task management for your entire company with Trello?
Yesterday, I logged back into to look at their new UI (which is very pretty) and it took me 5 minutes to figure out how to remove someone from the workspace
Trello doesn't have the design that Asana has, but it gets the done. You can also add task lists using markdown in Github issues, which is my personal favorite.