One thing to remember is that the p-value is a statistic---it takes your data and transforms it in a particular way. As a result, the p-value is a random variable. It has a mean and a variance. It is difficult to calculate the distribution of the p-value generally, but it is easy to calculate it under the null hypothesis: if the null hypothesis is true, the p-value has a uniform distribution from 0 to 1.
Since the p-value is random, it's possible that the two p-values that you are comparing are not statistically different. (Again, calculating p-values for differences in p-values is hard, typically requiring stronger assumptions than the problem gives. That two p-values are statistically the same, then, is hard to test.)
Andrew Gelman likes to say that "the difference between statistically significant and not is not statistically significant." In other words, 0.05 is not a magical cutoff and being below this value isn't necessarily different than being a little above due to the randomness of the p-value itself.
Another issue is that some procedures are more efficient than others. Typically, efficiency is gained by making additional assumptions about the data and using those assumptions to devise tweaks in the estimation procedure. It's possible that the two studies have the same estimate of the effect, but have different standard errors, leading to different p-values simply because one study makes additional assumptions that gets it lower standard errors.
Of real interest to you should be whether the two studies estimate the same effect. If the estimates are similar, most people would choose the more efficient/significant one unless they believe that the assumptions needed for the efficiency boost are not valid.
If the estimates of the effect itself are not similar, then you need to look at the two estimation procedures and determine which is more compelling---this isn't something that I can comment on without knowing the specifics.