Oh, where to start!
So Asana is a todo list. Everything is a task. Tasks are assigned. Tasks can have subtasks. They can also have comments, tags and attachments. The top-level organizing construct is the project; projects have tasks and also team members. Oh, and within a project, tasks have order (something which doesn't really work in practice).
So far so good. Most of the problem with Asana is in the UI. Asana is all about lists. Unlike Trello, for example, Asana only has one view: A vertical list of tasks. This list UI is very much like a classic outliner or even text editor, in that there's almost always a cursor that's editing something. You can use the keyboard to navigate around. This leads to some immediate weaknesses: Since there's usually a cursor, typing something will typically have the text land in some random, unpredictable place. Hitting Enter creates an empty task. So if you're not super careful, your lists will be littered with errand characters and empty tasks.
This information-dense view means there is never a place to "rest". I can't keep a single task open. There's a split screen that allows you to maximize a task, but it's not quite the same. So in Asana, you never feel like you're in a specific location. It's always the list. It's easy to get lost because there's little or no sense of hiearchy, even though the data model is hiearchical. This UI is doubly problematic because Asana is a heavy single-page app (it takes several seconds for it to "boot") that doesn't lend itself to being opened up in multiple tabs. So it's not very web-app-like.
Really, this kind of editor-like "live" UI seems more appropriate for personal todo lists, but not for collaborative ones. In my mind, a task is a kind of mini project, and a task with subtasks is a bigger kind of mini project (usually you use these for multi-step tasks like launches or big features), but Asana treats them all with equal lightweightness. In fact, in your "My Tasks" view (also a list), Asana will comingle subtasks and top-level tasks, making it confusing because subtasks don't carry their context with them — so "My Tasks" will list something like "Add CNAME", which on its own makes no sense, until you click on it and see that it's a subtask of "superproduct.com launch".
There are lots of minor flaws. The editor is horrible — pasting and undo/redo tend to make the cursor lose its place, there's basically no support for pasting code (no syntax highlighting), the keyboard shortcuts are non-standard and weird (they use tab (!) instead of alt or ctrl as a modifier), and you can't turn off the stupid rich text mode (no Markdown). As mentioned, startup time is bad, and the main page loads as a POST, so if you restart the browser, the browser needs to ask if you "want to submit this form", which is bizarre.
The more I talk about, the less I like it. Unfortunately, so far I haven't found any alternatives that look good enough to migrate.