Bill, as anyone will tell you, I'm batshit obsessed with project management tools, and I'm always trying to find the best one. The good news is that Asana and Trello are definitely two of the best solutions that exist: they're the most modern, the most ubiquitous when it comes to device support, and they're constantly being updated. They also look pretty, which matters.
Either Trello or Asana will work just fine for brainstorming. They both have the ability to list out ideas, to "heart" or like certain ideas (and vote on them), and to discuss things in comments. Asana also has a new "conversations" feature that allows you to discuss in long-form, almost like a chat or a forum, before actually adding things to your lists, though frankly I would just use Slack for that.
As a matter of fact, if all you need is brainstorming, Slack will often do the trick (and is an amazing tool in other respects, so you should get it regardless). Slack allows the team to add "reactions" to every message, and you can favorite messages that relate to cool ideas. Slack also makes it super easy to access files associated with brainstorming sessions. If you do this, I'd just create a channel called #brainstorming or something similar.
I would say the main difference between Asana and Trello is that Trello is for very visual people. Each board has literal cards that you can move around on the screen and attach to lists. Frankly, I think that metaphor is outdated and I prefer a more efficient view of things, which is why I use Asana. Asana also does a better job at remembering every little change and edit that is made to an item or project, which helps me keep track of when things happen.
Trello, for what it's worth, has a brand-new Slack integration which actually allows all your chatter in Slack to be attached to cards in Trello, meaning you can link your messy conversations about brainstorming (Slack) to the actual action items you arrive at (Trello), all in the same view. It's pretty awesome and is making me very jealous of Trello users. But Asana is just too streamlined for me to leave, and it also integrates with Slack.
Hope this helps.