I have a bit of both; Assos and Rapha. I like the styling of rapha and their shirts (sportswool) feel good quality. Some cycling friends have other rapha tops and once again, they look smart, are well made and if you can get them at a discount, might be worth it.
Assos, make excellent bib shorts and more technical tops; gilet, winter jackets etc. However, their normal shirts/jerseys are just pretty normal nylon based tops. I have one and whilst I wear it, I do not think it is any better or worse than many other in my drawer. My wife and I both have the Assos winter jacket - excellent kit, warm and wind resistant. Mine is hanging in my office right now.
So in conclusion, I personally do not think Assos summer weight shirts are anything special and I like the Rapha shirts more.