A website relaunch replaces the system underneath the site, not only the surface. The URL structure changes, the content model changes, and the way the site is edited changes with them. That makes it the way to replace an outdated system, and also the project in which search visibility built up over years can be lost quickly.
A redesign is the smaller intervention: it changes how the site looks and leaves addresses, structure and system in place. It carries little risk for your rankings, and it solves hardly any technical problem. If the site is slow because of its system, if nobody can edit their own text, or if plugin updates have become somebody’s recurring job, a redesign will not touch any of that. If none of those apply and the site simply looks dated, a redesign is the cheaper answer and the honest recommendation.
The rest of this page assumes the relaunch is the right call. What follows is what decides whether the new site keeps what the old one earned.