I've been told
that one of the considerations between SS and mild steel is fatigue life. A chassis used for a show car may not face this boundary, but a street-driven / racetrack driven car may. As eloquently stated above relative stiffness should not be too much of a factor.
The considerations in fatigue resistance include metal composition (e.g., mild steel, 312L, 304L, 3CR12, etc.), corrosion resistance (yes, Virginia stainless corrodes too under the correct circumstances), welding materials and methods and skill of the welder, joint stress (design) and I'm sure I've left out some others
Just like other design considerations for our favorite conveyance there are lots of factors and compromises involved.
Not simple, but it can still be worked out in a practical fashion...
Tom