Have been seriously looking into getting one of these but one thing which I don't quite understand - since we have a phone on us 90% of the time at home, what does this do which my phone can't? I already have multiple devices respond when I say "OK GOOGLE".
You ask a great question: What can Google Home do that my Android phone can't? Ultimately it is a lifestyle choice and hence these are my personal impressions. Google Assistant is built into both devices (although I believe some actions are on Home but not yet on the phone Google Assistant), but I find it more intuitive as the standalone Home speaker.
I for one do not wish to be tied to a phone screen. If you carry a phone in the house 90% of the time, it is often in your pocket where the response is muffled when you say: "OK Google....." It requires a separate action to pull the phone out of your pocket or leaving it permanently outside your pocket. If you use Assistant, you are trying to minimise your actions, not add to them. Having a conversation is the most natural thing I can think of doing with a smart assistant device. Being forced to take the phone out to hear a response reasonably well seems an unnecessary action when I do not need to do anything with Home other than speak. The quality of the microphones on Home that pick up your sound from across a room is excellent. Further, the quality of the response sound is much much better on Home than on my top of the line Android phone, the Pixel XL. Google Home has three quality speakers compared to a single speaker on the Pixel. So any music played sounds amazing, much better quality than my phone.
Google Assistant is super intelligent on Home. When the Pixel XL knows that it is near Google Home, it automatically allows Google Home to respond to everything, so you do not get your phone making any response or taking any action. What I find myself doing is not needing to carry my phone around the house. More and more actions are becoming more natural for me to perform on Home without lifting a finger! I do not feel as inclined to use Assistant on my phone with the frequency I use Home. For example, yes I can regulate my Nest Thermostat on my Pixel phone but I find it easier to adjust it by simply asking Home to do it. I find it easier to ask Home to play music because the quality of the sound is so outstanding.
And here is my final thought on why Home rather than GA on my Pixel. I find Home to be a great product for my entire family. Unlike my Pixel which I use exclusively and do not share with anyone else, Home's greatest selling point is the ability for group conversations to interact with it. Anyone in the room can ask Home a question or ask it to take an action. This distinction is clearly not lost on Google. Recently Google introduced the ability of Home to detect up to 6 different voices so that only you will be able to access private information on Home----ie, Home tied to your Gmail account will mean that only your voice can access your calendar, Gmails, etc., but anyone else can access non-private info on Home. Google introduced it first in the US and it is being launched in the UK very soon.
Hope this gives you some impressions as to why I chose Home in addition to having a Pixel phone.