If you are using Citi Premiermiles card, do note that you earn 1.2 Citi Mile/$. 1 Citi Mile = 1 KF Mile or 0.4 AsiaMiles.
All other cards that accept both will have the same exchange rate.
Mainly because most just think of Singapore Airlines by default. However, it is also true that you have the best availability of routes via Singapore Airlines, compared to oneWorld flights (eg, SIN-TPE direct is available via SQ, but for Asia Miles, you can take CX, but will need to transit through Hong Kong).
One thing to note also is that to redeem a Star Alliance (eg, Asiana, or ANA) flight using KF miles will cost you more miles. Eg. Using KF points, SIN-SFO return in Business class cost 136,000 miles on SQ, but 195,000 miles on any other Star Alliance partner. So, generally you would want to stick to SQ flights.
For oneWorld, as long as you stick to one carrier, the number of points required is the same, whether by CX or any of it's partners.
But to your original question, it's ultimately a question of how you fly. If you just buy the cheapest ticket with little regard to brand (or alliance) loyalty, then neither of the FFP will work, esp since deeply discounted tickets usually earn little or no miles (eg. some SQ tickets earn 10% miles only, hardly worth the trouble).
As has been mentioned a number of times, award tickets (even accumulated with bums in seats) are really only worthwhile if you would like to take premium seats (Business & First), and particularly for long haul flights. If you are happy in Economy class in any airline, you can find discounted fairs on some airline at the price that SQ or CX would charge you for the award ticket's tax & surcharges.