This discussion comes up in every WordPress project sooner or later. The client asks which SEO plugin they should use, and the answer depends more on circumstances than on the plugins themselves.
What they do (and don't do)
Both analyze your content, generate XML sitemaps, handle schema markup, and give you control over meta titles and descriptions. Neither can compensate for poor content or a weak link profile. SEO plugins help with the technical side, not with relevance.
The free version
This is where they differ. Rank Math gives you schema markup, redirect manager, 404 monitoring, and basic keyword tracking without paying. Yoast locks most of that behind Pro (99 USD per year). If budget matters, the choice is obvious.
Ease of use
Yoast's setup wizard is smoother for beginners. The traffic light system (red, yellow, green) is intuitive if you don't have SEO experience. Rank Math has more settings exposed directly, which can feel overwhelming if you don't know what you're looking for.
On the other hand: if you're experienced with SEO, you'll appreciate Rank Math's transparency. You see exactly what's happening without clicking through premium upgrade pages.
Performance
Rank Math weighs marginally less. Not enough to notice day to day, but measurable running repeated Lighthouse tests. The difference is maybe 20 ms. Not a decisive argument.
In practice
We install Rank Math in new projects when the client has no preference. In existing projects with Yoast, we rarely switch. Migrating between SEO plugins works, but it creates extra work (especially with redirects and custom schema) that's seldom justified.
Pick one and learn it properly. A half-configured Rank Math is worse than a thorough Yoast setup. And vice versa.