Just now I'm enjoying Haslinger. Superb in every way but with barely scented. A refreshing change from even quality soaps which have concoctions of aromas.
My preference for scented or unscented soap is dictated both by what my skin is doing that day, as well as my fragrance choice. I have some fragrances that pair beautifully with various shaving soaps. For example, His Majesty The Oud EDP and Tobacco Oud EDP both pair well with Ariana and Evans Incense and Oud, in my opinion. I think Martin de Candre Fougere pairs well with Fougere D'Argent EDP. Other soaps and fragrances in my collection have such idiosyncratic olfactory profiles that I have to choose between unscented soap and fragrance or no fragrance and scented soap.
In answer to another piece of your question, I find unscented soaps, like their scented brethren, vary widely in emollients, lubricity, and post-shave face feel. Having the right choice for whatever state of idiotic flux my facial skin is in at any given moment seems less like a luxury and more like prudence.
If different soaps with fragrances behave differently, and they do, then different unscented soaps can/will also behave differently.
Buy a couple/few, try 'em out.
Just about every soap smells like something though. Unscented to me is just "un-perfumed".
AOS store gave me some un scented cream samples years ago. I used mine and the wife. I was *** there is no real smell, the wife however said it was nice after use.
But I think a lot of the reason I use soaps now. They don’t smell like the shaving cream or gels I was using for years. They are not long lasting smells either. YMMV