The main change for those on iOS is that there'll be fewer navigation tabs -- instead of five, there'll now be four. By default, those four tabs are for Home (your timeline), Search, Notifications (or Mentions) and Direct Messages. To access your profile, lists, settings and other accounts, simply swipe right to see a new side navigation menu. Those on Android are likely already familiar with this, as they saw this design change last year. But now iOS users get to enjoy it too.
Another change to the iOS app is that links to articles and websites will now open in Safari's viewer rather than the in-app web viewer. According to a Twitter spokesperson, this is so you can easily access accounts on websites you're already signed into, like if you have a New York Times subscription for example. Using the Safari view also gives you the option of using Safari Reader and you can get to use private browsing mode too if you wish.
There'll also be design changes that go beyond just the iOS app. For one thing, the reply icon has been changed from a curved arrow to a speech bubble. Apparently this is because some people thought the curved arrow meant "delete" or "go back to a previous page." The icons underneath each tweet are lighter as well. Typography will be more consistent, headlines will be bolder and profile photos will now be rounded instead of square. The reason for the round profile photos is to better distinguish them from in-line images in tweets. These design changes will be on the iOS app, Twitter.com, Twitter Lite, the Android app, as well as TweetDeck.
A particularly interesting change is that reply, retweet and like counts underneath tweets will now update in real-time right in front of your eyes. So if a tweet goes particularly viral, you'll see the retweet and like count on it go up rapidly without you having to refresh your timeline. This particular design change will only be on TweetDeck and the mobile apps; it won't be on Twitter.com or Twitter Lite.
Twitter says it'll listen to feedback on these changes and it'll have more design updates soon. In the meantime, we'll continue to hold our breath for that much-requested edit button.