Newton designer Umesh Gopinath wrote a pretty interesting blog post about the inspiration for today's update, noting that Gmail's conversation view all but obviated the need for the traditional "sent" folder. You could see your send message alongside the responses in a few that's a lot closer to a chat conversation than traditional email.
As of today, Newton will show sent messages in the inbox, just like other conversations. Indeed, Newton's view now works a lot like a messaging app. When you send a response to an email, that conversation moves to the top of your inbox; if you start a new conversation, that message will be at the top of your inbox until something else pushes it down. If you send a ton of messages, this might get annoying, but you can always archive those messages. If, on the other hand, you want to easily see messages you've sent and follow-up on them, this new style of inbox may work for you. Of course, it's not a requirement -- you can turn this new feature off if you prefer the way Newton worked before.
I tried an early version the Newton update, but unfortunately our corporate email doesn't work with apps like this one. I did hook up my personal email, but apparently I don't have enough friends for me to really judge its usefulness. That said, I can imagine that people who send a lot of emails might appreciate being able to easily and quickly see what messages they've sent that haven't had a reply. It's perhaps not quite the email revolution that Newton promises, but it's fun to see a company try and take down something that's been around as long as the sent email folder.