I think it really depends on what mechs you play and their level.

Both are good for dealing damage but if you get C10 and can't equip both becuase no mech you own has an energy 20 capacity it is kind of a waste in the short term.... although long term it will be better.

I personally always prefer what I will use long-term and would rather put resources into that becuase once you get it and can equip both there is a huge difference between C8 and C10.  It will do about 6k damage more per weapon/mag so with both equiped it is like having a 3rd weapon with the extra damage.

I have personally found it works best on tanks like gaurdian that can just disable and shred you quickly.

You get two Carbine 8's and only one C10 and only one C12.  I unlocked the 10 because I did not know there was only one in the set (similar to long arm 10).  Although I would have waited to unlock the 8's, the 10 has been a good weapon for my Zephyr along with pulse cannon 6.   

You don't get two C10's, only one until you unlock the duplicate.

Sorry if I didn't specifiy that but it is why I stated it was a long-term investment.  I ran into something similar recently.  On my redox I had mortar 12 and Mortar 10.  I put all the resources into Mortar 12.  now that it is leveled up to 7 I get a second copy.  The short-term benefit of putting resources into Mortar 10 for me was just not worth it.

You will still need a Carbine 8 for the mechs with 18 power so keep that in mind.  I run a Carbine 10/8 on my Killshot, Ares, and Jugg regularly.  Other 24 pwr mechs like the Panther, Redox, and Gaurdian will use a Carbine 12/12.  So basically, IMO, all of them are important to invest in.