We've been able to talk to our Android phones in a conversational manner for years thanks to the power of Google search. However, Google has been focusing more on voice over the years with features like the "OK Google" commands. Then at Google I/O, the company started talking about Assistant, but it neglected to really explain what Assistant is and how it's different from the Google search features you already have on your phone. Let's see if we can work it all out.
At its heart, Assistant is an impressive artificial intelligence engine. It's like Google search, but more powerful and aware of context. Google's voice and text search features in the past have always been built into the Google Search app on Android. However, Assistant is designed to be integrated into more places, and it will be explicitly labeled as Assistant instead of just being some amorphous "Google" thing that can go by any number of names depending on the context (eg, Google Now, voice actions, and so on).
In the new Allo chat app, Assistant is what powers the Google chatbot. On the Pixel (and maybe more Android phones in the future), Assistant is the omnipresent voice-activated search tool. In Google Home, Assistant can do voice searches and control connected smart home devices like Hue lights and the Chromecast.
The way Google Assistant and all of Google's other voice tools is really unchanged. Your words are transcribed locally on the device, then uploaded as text snippets. That's faster than streaming the raw audio to be processed in the cloud. That's why Assistant in its chatbot form and conversational voice form work in mostly the same fashion. So, when you hear about Google Assistant, it's not really a replacement for Google voice search or OK Google—it's more of an evolution.