I don't have any specific task or algorithm in mind, so depending on how they were tested – Is there any research which shows just how the D-Wave Two computer was faster (in terms of computation performance) than its predecessor (D-Wave One)?
As Troyer and Lidar saw no speed increase with the D-Wave 1 compared to classical computers, the D-Wave 2 benchmark figure reported in 2013 of 3600 times as fast as CPLEX (the best algorithm on a conventional machine) suggests the D-Wave 2 is 3600 times as fast as the D-Wave 1.
- the results are in a pretty restricted set of parameters, so this may not be relevant for other parameters. (as an example, the benchmark figures for the D-Wave 2000Q only take constant factor performance gains into account)
- the configuration of the CPLEX may not compare directly to the classical computers used to benchmark the D-Wave 1
As far as I know the closest answer to your question for applications is given in the recent (still unpublished) work presented at the March meeting by Bibek Pokharel, where he compares graph 3-coloring instances on D-Wave Two, D-Wave 2X and D-Wave 2000Q, all other things staying reasonably equal. The short answer is that all the performance increase is essentially due to the possibility to run single anneals at shorter anneal-time. (e.g. 1$\mu$s instead of 5$\mu$s gives indeed about 5X of performance increase, in terms of time-to-solution (TTS) metric. With respect to 20$\mu$s of D-Wave Two the scaling is different).
I can also spoil that from D-Wave Two and D-Wave 2000Q on Sherrington-Kirkpatrick instances we observed no substantial improvement as well. Results will be published soon in collaborations with Stanford.