Seriously, different frame designs do different things.

On road bikes, the frame geometry determines the handling -- very sharp for crits, slacker angles for road racing, laid back and easy for touring and randonneurs. Look at TdF and you will see the same rider user different bikes for different stages -- time trials, the flats and the mountains.

With MTBs, it may be full-suspension Y-frame (or whatever) for downhill where weight is faster; something different for X-country. KHS, I think, even makes a Cro-Mo bike that relies on the flex in the chain-stays, to provide suspension (the seat stays meet at a device atop the seat tube). And what about BMXes?

You *can* make a bike designed for one use do something else, but it could be a challenge to ride comfortably -- a BMX as a touring bike? I don't think so.

So, stripping away the science of materials, yes, frame does make a difference.

However, custom frame builders know their craft, and think nothing of *mixing* tube materials to get the best *perceived* ride quality. Of course, they concentrate on the most appropriate dimensions for the rider as the starting point. Again another point -- the efficiencies of tig/mig welding mean lots and lots of bikes coming out one end of a factory that require compromise in fitting for the end-user. And that's OK. Tig/mig is a money matter, related to efficiency, not necessarily (and I say that advisedly) an improvement over the traditional methods.

To do a test on whether one material or another is better, diamond-frame bikes of exactly the same dimensions and angles, including fork, would need to be built and "blind" tested. IIRC, a magazine had this done, and the difference in ride quality between three (?) different metals types was negligible.

And, if you *really* want a definitive answer on whether frame design does make a difference (as opposed to material), talk to a recumbent rider!