The biggest drawback to how MyMail serves up your accounts is the lack of a unified inbox. In other words, you can't view all of your unread messages across multiple accounts in a single view. Instead, you're forced to bounce back and forth between accounts to view your various inboxes, taking up time and potentially annoying you to no end (depending on how many accounts you manage).
Creating a My.com email account is useful if you want a private email that only exists on your phone, but it has its drawbacks. Most notably is that you can't view your emails on a computer, so if your phone dies from a drained battery, there's no way to retrieve those emails until you recharge it. Managing your My.com email on a phone might be acceptable for some, but I imagine it wouldn't be ideal for those who receive large amounts of mail on a daily basis.
No real surprising features
The one special feature MyMail offers is the option to customize message notifications based on sender, social network, or other online services. The option makes sense for its own email offering, but not for other services. Labels and filters are standard features found in the likes of Yahoo! and Gmail and take care of customized notifications, making the MyMail features a moot point.
Dealing with attachments, both sending and receiving, defaults to locally stored files (in the case of iOS, it only allows for sending photos, not actual documents or additional files). Here, direct integration with the likes of Dropbox or Google Drive would be beneficial. You can attach files from either service using the Android app, but the integration is done through Android itself, not the MyMail app.
Indeed, managing multiple e-mail accounts is a frustration point for many. But I fail to see how MyMail puts an end to the frustration and I think at the very least, it should offer a unified inbox. The stock apps on Android and iOS both offer similar feature sets as MyMail, giving MyMail no real breakout feature. Granted, it offers a My.com account complete with as much storage as you need, but the "mobile-only" approach is prohibitive. Do you really want to be limited to composing messages, and managing your inbox from your smartphone or tablet? I know I don't.
Still, MyMail is a breath of fresh air with a clean interface and worthy of a try if you're bored of the standard email apps on your platform of choice.