A router is a device that "routes" network traffic. Devices have addresses called IP addresses, or you can have a domain name like reddit.com that maps to a device's IP address. When you send a message over a network you send the data and the address. Your computer is connected to the router, which is then connected to other computers on the same network or on the internet. The internet is in some sense just a network of routers and computers attached to them. A router is the device that gets the message to the right place. (It's a little more complex, there are switches, DNS networks of networks and more).

A router is also just a computer. It has a processor and memory and all that. They usually have specialized hardware and operating systems that specialize in "routing" stuff, but they are fundamentally just computers.

A home automation hub is a computer that is used for the central communication of controlled devices. When you hook up a controlled light, you would "network" it to the home automation hub. This way the home automation hub can talk to the light and then turn it off or on. You would also usually connect some kind of home automation application to the hub. This way when you send the "turn off the living room light" message to the hub from your phone or something, it is networked to the light and can then send the message to the light.

Usually a home automation hub will support additional hardware that computers don't typically have like different frequencies of radio transceivers, IR ports, relay ports, IO ports, RS232 ports and more. Unfortunately the google device only supports a radio transceiver for Zigbee and ethernet and from just looking at the picture of the device there's no way the hardware is upgradable.

So a home automation hub is just a computer that runs some software that talks to devices and it may support special hardware. Usually the "talking to devices" is just routing messages around, so if you hit a button on your phone it will send the appropriate message to the appropriate device. You can usually also run local services on the hub, for example timers so you turn off all your lights at a certain time. From the developer docs on this particular device, it looks like the "home automation" part is basically just a web server that translates how different devices talk and passes the messages around and device drivers for zigbee.

So a "smart home hub and a combination router" is computer, running 2 applications, the router part(which is actually probably mostly part of the operating system) and a webserver application for home automation.