Essentially, 2d6 will give you a higher likelihood of rolling a mid-range number on any given roll, and 2d6 is more likely to give you somewhere between about 4-9 damage on a roll, whereas a 1d12 will give you an even chance to earn any number, and tends to be more unpredictable. You'll end up with more large damage hits, but also with more small damage hits. Basically if you want more consistent, reliable damage, use 2d6, but if you're willing to risk the higher chance of a lower roll for better chance at higher rolls, use the 1d12. Mostly a personal choice based on your own risk tolerance. Even better, having both available can allow you to use whichever suits the situation you're in.

Code
                                            For 2d6                                                               For 1d12
Total of dice |  Permutations                              | Probability |                            | Total of dice |  Probability |

      1             0                                               0%                                       1              8.33%
      2             1 (1,1)                                      2.77%                                       2              8.33%
      3             2 (1,2)(2,1)                                 5.55%                                       3              8.33%
      4             3 (1,3)(3,1)(2,2)                            8.33%                                       4              8.33%
      5             4 (1,4)(4,1)(2,3)(3,2)                      11.11%                                       5              8.33%
      6             5 (1,5)(5,1)(4,2)(2,4)(3,3)                 13.88%                                       6              8.33%
      7             6 (1,6)(6,1)(5,2)(2,5)(3,4)(4,3)            16.66%                                       7              8.33%
      8             5 (2,6)(6,2)(3,5)(5,3)(4,4)                 13.88%                                       8              8.33%
      9             4 (3,6)(6,3)(4,5)(5,4)                      11.11%                                       9              8.33%
     10             3 (4,6)(6,4)(5,5)                            8.33%                                      10              8.33%
     11             2 (6,5)(5,6)                                 5.55%                                      11              8.33%
     12             1 (6,6)                                      2.77%                                      12              8.33%


I added that to show a breakdown, but in D&D you shouldn't really be fishing for a certain number, so the individual percentages are not really useful on their own, and I think it's better to look at the odds of getting within a range of damage. Broken down into four ranges:


Code
     2d6              1d12
Roll | Range      Roll | Range

1-3:    8.33%      1-3:   25% 
4-6:   33.33%      4-6:   25%
7-9:   41.66%      7-9:   25%
10-12: 16.66%      10-12: 25%


Statistically, over many many many rolls, the 2d6 will typically do more total damage(about 7%), but that doesn't matter much when a battle is hinging on a single roll and where the line between success and failure lies.