Hi Vandal,
The sound character of both is such that the midrange is the star of the show. Both provide crystal clear clarity, which when paired with the wrong amp, may come across as etched or tipped up.
The major difference between the two as I hear things is balance, top to bottom range-wise. Where the 1Ks’ mid-range is quite sweet, the 5Ks have more and better delineated bass and a high end that is a bit smoother. Thus, I could see some see the 1Ks midrange as better when, in reality the 5Ks' mids are equal or better but just not as prominent. The 5Ks also provide a slightly larger soundstage as well.
To me, both cans work best with tubes. That said, the 5Ks work better with a much wider variety of amps and do well with SS also. Because of this, I believe that the 5Ks are without a doubt the better (overall) set of cans.
Until recently, the 5Ks where paired with my Opera in my work office and I thought that they worked just fine together. My 702s have taken their place as my 5Ks are now at home doing time with the Yammy and Cayin.
With that said, I still prefer the 1Ks with my Yammy. The two were literally made for each other as Yamamoto Sound Craft provides the cups for the 1Ks.
Hope this helps a bit.
Cheers,
Garry