@Glenn:
Check this post from Dabbletron. It’s a detailed comparison of AstroPrint Vs. Octoprint.
To answer your questions…
Astroprint seems to derive from Octoprint, but does NOT offer Internet access for free, only local…Is this correct?
No, it’s not correct. Astroprint is 100% free.
Astroprint seems to derive from Octoprint, but does NOT offer Internet access for free, only local…
Again, Incorrect. I am curious, where are you getting this information? Can you link us?
I like the interface of Astroprint better, but making it a paid subscription seems strange to me, when you own the printers yourself, and the hardware Astroprint is installed on.
When AstroPrint started LONG time ago, there was a subscription tier – perhaps thats the cause of all this confusion. Again, it’s 100% Free. With that said, we do have a few premium paid APPS on our APP store – but these are all optional. Similar to Android – in that, you can use Android for free, but you can also purchase premium apps from Google Play. The AstroBox also costs money, but that’s if you don’t want to make your own. You can make one for free with a raspberry pi and our images.
Also I see that the Astroprint is using an externally hosted cloud system, is this really necessary?
Yes. It is necessary. Because AstroPrint is NOT OctoPrint. AstroPrint is aimed at a very particular user (non technical/geeky general consumer who needs a plug and play solution for their 3D Printer). Advanced users still have access to settings under the hood (I.e: Advanced Slicer Settings) – but most of that is not hugely prominent in the Interface.
AstroPrint also has a few more key components that makes it essential to be on the cloud: an APP Store (which eventually, any developer can write 3D Printing related APPS for), Automatic security and slicer updates that is done server side, and the ability for 3D Printer Manufacturers to control optimal settings for their 3D Printer. For example, If you have a PrintrBot Printer, and you are using AstroPrint, the optimum default settings for the PrintrBot are pulled from the cloud and has been confirmed by the Printrbot manufacturer themselves.
Isn’t it possible to run the whole system from a home computer/server instead?
Yes and No. You can run your AstroBox completely locally, but you’ll not be utilizing the cloud component – which, in my opinion defeats the purpose of AstroPrint.
I can understand the confusion. Let me know if I could clarify any more confusing parts 
Also, here are a few more reasons why AstroPrint is different from OctoPrint:
1.) Faster Slicing: Once the file is up in the cloud you can re-slice the file (for higher/lower quality) with any combination of settings you’d like as preferences change. Why would someone want to do this? Many people end up tweaking and experimenting with various settings that require re-slicing. AstroPrint has powerful dedicated machines reserved JUST for high speed slicing, machines that are many many times faster and powerful than home computers or laptops. You can also do this from anywhere around the world without the need for software
2.) Custom Manufacturer Presets: The AstroPrint slicer has fine tuned settings for most printer manufacturers PRE-BUILT into the system so that you don’t have to worry about googling and tweaking settings to match the best configuration. Not only that, AstroPrint works with manufacturers directly in obtaining these settings and allow them to tweak settings directly. Also, if these optimal settings change in the future, no worries, AstroPrint automatically updates those settings so you can focus on printing cool stuff.
3.) Remote File Management: Once the file is in the cloud, you can slice the file remotely from anywhere. As a bonus, you can also monitor the print live (with a camera) and also start/stop the print if necessary.
4.) Simpler to use: It’s super accessible for non-technical users. Advanced settings are still accessible for advanced users, but there’s nothing simpler than AstroPrint in terms of UI/UX and it will only get easier to use.
5.) Automatic Updates: Software & security updates are done automatically. The average user does not have time to worry about software/slicer updates & security issues. When critical updates are available for the slicer, AstroPrint takes care of all of that so users can focus on printing stuff.
6.) Plug & Play Slicing Engines: The slicing engine is modular. Meaning, with a click of a button you can use Cura or Slic3r and in the future, other slicing engines can be added easily like lego blocks.
7.) APP STORE: AstroPrint has an APP STORE. When the API is open, anyone can submit apps to the APP store that interacts with the AstroPrint cloud platform. This gives you the option of enhancing your 3D Printer at will, with one click. Work in a library and want an app that will help you manage users and access to a single 3D Printer? There’s an app for that. Need a premium app with a cutlery kit designed by a french chef? There’s an app for that. You get the idea.
Pick your poison though, can’t really go wrong with either 
