One issue is that chat is now hidden as soon as you start any call, be it audio or video. I could possibly see where a video caller might want the full window to be consumed by video, but I certainly don't need a huge avatar on my screen during an audio call.
Maybe the folks at Skype are trying to give our eyes a break from all that blue, but chances are in both scenarios I'll have some links to share or text to type, and for that I need the now-hidden chat. Don't worry, it's easy to re-enable.
The new Skype makes chat invisible during calls
Re-enabling chat inside a Skype call is simple: just click the chat bubble button in the upper right of the call window. If you don't see the button, don't worry: it will appear as soon as you move your mouse over the window.
Changing Chat Location
You'll likely see that the chat now appears to the right of your video. I suppose this might look good on a small laptop screen but I prefer it below the chat because I like the video oriented near my camera, which is at the top of my Mac's screen (on your landscape-oriented iPhone or iPad the camera would be centered, making this design choice possibly work better there).
When enabled, Skype's chat often appears to the right of the video.
The location of the chat is determined by the width of the Skype video window. If it's 720 pixels or less it appears at the bottom. More than 720 pixels of window-width will make the Skype chat appear on the right. If you have your Skype friends list on the left (not shown in these screenshots) you can add about 200 pixels to those numbers, depending upon how big you keep your friends list.
Narrowing your Skype window will cause the chat to move below the video.
Skype updates have been coming pretty quickly lately, so it's possible any of this could change in the next month or two. For now, you know what to do.