OK, so the most obvious question is likely something related to why there are two extremely similar functions that essentially do precisely the same thing. The answer is that while they are similar and do similar things, the way in which these functions go about what they do is fairly different. The order of the resulting table rows returned by these functions provides a hint that they are operating somewhat differently within the bowels of DAX.
Without going into excruciating detail regarding the internals of DAX calculations, the best way to describe the differences is to understand that within name/expression pairs of a SUMMARIZE function, the expression portion must always refer to an actual table name within the data model, while within the name/expression pairs of a GROUPBY function, the expression portion must always refer to CURRENTGROUP. This means that if you were to nest two SUMMARIZE functions, you could not refer to the table or columns created by the first SUMMARIZE function within the second (nested) SUMMARIZE function.
However, with GROUPBY, it would be possible to refer to a column created by the first GROUPBY function within the second (nested) GROUPBY function. This can be fairly powerful under the right circumstances. Conversely, because the SUMMARIZE expressions do not need to refer to CURRENTGROUP, this provides a certain flexibility that is not possible when using GROUPBY. For example, if you were to change the last parameter of each formula, within SUMMARIZE, as shown in the following line of code:
DAY(MINX('R05_Table',[Value]))This would work and you would return just the day of the first date of the weekday within each month instead of the full date. However, you would receive an error in the GROUPBY formula because the specified expression does not aggregate over the current group.