Which is faster, to return ajax in JSON and then process JSON response to render the html, or just have the Ajax response the raw html in a bunch of
As already mentioned, that depends. From a server side point of view it makes a lot of sense to let the client generate the HTML because just serializing JSON is faster and takes a lot of strain off the server because it doesn't have to deal with all the HTML generation. An additional benefit is, that you offer an API when returning JSON that can be used for more than just outputting HTML.
If you want to take the work off the client it makes sense to generate the HTML on the server side.
In the end the speed of it depends a lot on the technologies used. Both ways can perform extremely well but when done wrong either one will be slow.
Depends. In both cases, the server is simply returning a response with text. If the JSON version of the response requires more characters than the HTML version, that response will take longer to be transmitted back to the client, and vice versa.
But of course there is also the server-side script which must do its work. Perhaps in your case generating JSON is faster than HTML from your server-side script. No way for me to know.
And then there is the client-side processing. You'd have to parse the response to turn it into a true object, and then you'd need to iterate over the resulting object in order to generate the HTML. This will definitely take longer than just taking an HTML response and injecting it into the DOM.
However, I doubt that the performance difference will be noticeable, meaning that your decision about providing a JSON response vs. HTML response should be based on other factors.