Both are distinct with regard to their probability of likelyhood: as its name indicates, an "upper bound" estimate (UBE) is a "higher" value that is still consider reasonable with regard to its probability of occuring (e.g. 90 or 95th percentile value), whereas a most likelyhood estimate is the most probable overall, thus not the most probable "high" value like the UBE. With limited number of data, the MLE is a safer bet as to its representativeness of the available data, since upper bound values are often more uncertain when considering the tails of a probabilistic distributions of values; often, you need more data to be able to determine a likely robust UBE in any given distribution. However, from a risk assessment perspective, UBE are considered more conservative, therefore they may be chosen even if the may be more uncertain. Thus, your choice will depend of your objective: being more "safe" or being more "likely representative". 

I agree with the preceding points by Mathieu and would like to add a caution. Sometimes MLE can be badly biased for small sample sizes. A way to get a handle on this is to make a simulation - 1. come up with with an imaginary final result (an arbitrary example of the quantity you are trying to estimate whether by UBE/MLE),  2. work back to produce some random example estimates of the data, 3. then apply the MLE analysis. If you do this a few times, and compare its behaviour corresponding to larger data sets it might give you a feel for how much bias there could be for small samples.

a kind of objection

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...