Since you don't seem to be much concerned with UI/UX and want to know more about SEO, I'd suggest to go with the client side. Anything that's dynamically loaded after user's input won't be visible to web crawlers.
However, another approach would be to make it work both ways - so that by visiting a specific URL (site.com/search?q=something
) you get the page fully rendered from the server side, while you're still able to make another search that will happen at the client side. You'd still have a little trouble indexing all the relevant searches, but perhaps you could track the last x searches and show them somewhere on the page, with links to full server-side rendered search pages, like the one I mentioned above. You can even make those dynamic calls not only change the content of the page, but the URL hash in the browser's address bar as well (see here).
That way you'd provide users with a nice user interface/experience, while still doing a very nice SEO job since the crawlers would be able to index the links from the list of last searches.
So, to directly answer your question: client-side vs. server-side page rendering - huge SEO difference