Hi, the mean and the mode of the posterior distribution are just two measures of the central tendency of the distribution. If the posterior distribution of a parameter is normal, the mean, the mode and the median will be the same. However, typically the exact distribution of the posterior is not known. In this case you have to pick a measure. Usually with irregular posteriors I would go with the mode. To see the problem consider a binomially distributed posterior with p=0.6. In this case, 60% of the draws will be 1 and 40% will be 0. The mode will be 1 and the mean will be 0.6. I would argue that the mode represents the distribution better than the mean (particularly as the latter is not guaranteed to fall on a value actually obtained in the sample).
However, I am sure you can construct cases where the mean is a better measure. I would recommend plotting the posterior distribution and see where the mean and the mode are for the parameters and then choose the one that better represents the sample.
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