It was fairly easy to embed playlists from Spotify in blogs and websites. The process isn't quite as easy for Apple Music. Nor will the embedded playlist play the entire song, even if you are listening on your mobile device with an Apple Music subscription. You will get 30 second clips, on a computer or mobile device.
1. Go to https://playlists.applemusic.com/
2. Enter the url of your playlist. You can find this url by clicking on the share button on your playlist and copy the link (or send link as a text message to see link).
3. Copy the link to the playlists.applemusic.com site.
4. The embed code will be generated.
5. Copy the embed code the html for your blog/website.
This is what it will look like:
You can also embed a player for any song/video on Apple Music that looks like this:
If you would like more details, 9to5 Mac has more information.