In the new desktop app certain sections have been relocated.
You can find everything that was under Browse by going to the new Search tab in the left sidebar.
You can read this support article on how to use Radio
On the top row on the Home tab page in the new desktop app there should be a "recently played" section and you can click to "view all". It doesn't show up as an item in the left sidebar like in the old desktop app. If you are affected by the bug listed here you won't see this section until this is fixed.
Made for you:
To get to Made for you section click on the search tab in the left sidebar and then find the "made for you" tile.
Liked Songs are available as a pinned item in the left sidebar. You can also get to it by first clicking on the "Your Library" tab in the left sidebar and then clicking on the big "Liked Songs" tile.
Albums, Artists, Podcasts:
You can get to these by first clicking on the "Your Library" tab in the left sidebar and then at the top you will see "Playlists, Podcasts, Artists, and Albums sections.
You can read more info about the new desktop app in this news article:
The desktop team also has a blog post in the community about it:
There is also an FAQ about the new desktop app:
FAQ: The New Spotify App for PC and Mac