There's been a lot of action this year on new email clients for iOS and Android, with a good dozen startups trying to fix the standard Apple Mail and Google Email clients. There's a problem: Mobile mail is not broken. Although some tools have an interesting abaility or two, overall they are less capable and harder to use than the tools they seek to replace.
A strong contingent out there believes email is bad and should be replaced with social-style messaging systems. They're nuts. The mobile mail folks don't quite share that view, but they believe that email clients should do more than email, which is also why so many are hard to use.
I can think of good reasons to use a different email client in iOS or Android. One: Your company wants assured separation of work mail from personal mail, enforced by requiring a separate client such as the one installed by your MDM provider or even Microsoft's OWA app for Office 365 users.
Another reason is that you are a die-hard Gmail user and want to use Google's Gmail Web interface on your mobile device. It's not a great reason, but it's a real one.
The various mobile email clients I've tried -- OWA, Gmail, Mail+, Mailbox, and Boxer -- don't support all the email account types that Apple Mail or Google Email do. That especially matters for business users. For example, OWA supports only Office 365, but not on-premises Exchange. The free Gmail supports only Gmail. Neither Mail+ nor the $9.99 Boxer support Office 365, though they support on-premises Exchange (you need the $5.99 Mail+ for Outlook to get Exchange support), nor do they support POP accounts. The free Mailbox supports only Gmail and iCloud IMAP.
I'm also skeptical of the free email client apps. Developers need apps to make money, and free apps typically do so by analyzing your communications and personal data for sale to advertisers. I don't want either my personal or business email used that way. That's publicly the point behind Google's Gmail. The other approach to free apps is to get a big customer base, then force a switch to a paid version.
Still, there are some features that you might wish Apple Mail or Google Email supported.
For example, OWA lets you ignore a thread, so all the replies don't disturb you. You can also create a reminder from email text, similar to how Apple Mail lets you add contacts and calendar events from text in messages. Additionally, OWA lets you see your (Office 365 only) calendars and contacts, but switching among them is no different than switching among separate apps; there's no advantage here other than the need to keep work stuff truly separated. The ability to set a password to open the app is a nice feature, though.