This is a great way to formulate the difference. In my mind, Google does search, which is not the same as synthesis. ChatGPT is a synthesis engine. I think there is room for engines like ChatGPT to also return links to the training documents that were most relevant to producing an answer. At that point, it would be doing search and synthesis at the same time.

I am not sure if neural networks have been trained to do this yet, but it would be very cool to see a network that produces both generated output and the most relevant input data that led to that output, by somehow keeping track of the influence certain inputs have on various learned features and internal structure of the network. I think of this as an index.