Support

I know about the issue and I am already in touch with Last.fm on this. The temporal solution is to update the app to version 3.1.1 which should be available shortly.

You can download NepTunes for free but some features requires NepTunes PRO.
You can unlock it with a yearly subscription (with a free one month trial) or a one-time purchase.
Why subscription?
Great question. The software is never "done". And it takes substantial amount of time to make something like NepTunes to be as reliable and polished. The app has to deal with 4 different web APIs, two different music players and their APIs, bugs in them, tracks with missing metadata etc. It took me over 6 months, working almost every day, to create the 2.0 release and it was missing a lot of features!
Btw, I'm also not a fan of subscriptions… I don't have anything against this model – it's just because most of subscriptions are overpriced. And I don't want to do the same thing that would personally pissed me off. And few bucks yearly for a non essential features is a cheap subscription. I will have stable stream of (small) revenue, and you, as the user, will know that you didn't pay 30 bucks for an app that will be killed 6 months later.

To fix it you can:
a) Update the app to version 2.3.2 or later
or
b) Delete the recently played tracks:

• 1. Open Finder and press CMD+Shift+G (or click on the "Go" menu and choose the "Go to Folder" command)
• 2. Copy and paste this path:
  ~/Library/Containers/pl.micropixels.NepTunes/Data/Library/Application Support and press the return key
• 3. Delete the NepTunes folder
• 4. Launch the app

1. 1. Quit the app
2. 2. Open Finder and press CMD+Shift+G (or click on the "Go" menu and choose the "Go to Folder" command)
3. 3. Copy and paste this path:
   ~/Library/Containers/pl.micropixels.NepTunes/Data/Library/Application Support
4. 4. Delete the NepTunes folder
5. 5. Launch the app

Sure thing.
Please check out the roadmap – maybe it's already there.

It looks that Apple Music app has a bug and when the track is played from Apple Music catalogue (so from the search results, the browse section and the listen now section) the player won't provide the album artist for NepTunes. I've already reported it to Apple (rdar://FB9919534)

First, verify that the app has correct permissions and has access to Apple Music/Spotify. Go to System Preferences -> Security & Privacy -> Privacy -> Automation. There should be NepTunes with Music.app and/or Spotify.app selected. If there is no NepTunes at all, restart your computer, launch NepTunes and from the menu choose the More submenu, click on the Quick Start Guide… and click the "Grant Access to" button.
If the app has correct permissions check out the Last.fm status.