The Echo Dot is exactly the same as the full Echo with slightly less sound quality. All other functionality is the same. All you need to use it is WiFi. You don't need an Amazon Prime account to use it but if you have Prime you then have access to all Prime music. Even without that, you can create a free Pandora account and get music through that. If you have Kindle books, it is able to read those and they actually sound pretty good - much better than other computer-generated voices. If you have an Audible account, it can play your Audible books. It's search engine is pretty good, but not on par with Google.
The newer Google Home is similar to the Echo, but wouldn't have access to your Kindle or Audible libraries. It does have the better Google search engine.
So, if you have an investment in Kindle books and/or Audible audio books, or Amazon Prime, I would go with the Echo. If you don't have those investments, I'd go with the Google Home to get access to the better search engine.