"early in the morning" and " in the early morning"

May we assume the common bits make no difference and you're actually asking for a comparison of 'in the early morning' and 'early in the morning'?

No. There is no significant difference.

In context, there is very little in the way of practical differences in semantic meaning here.

Linguistically, this demonstrates an interesting phenomenon, which might be underlying your intuition that there is a difference between these two sentences (if indeed you get this intuition). This is an example of the difference in modifier scope (a term borrowed from logical quantification). Essentially, what this means is that the modifier early is taking scope over two different parts (or constituents) of the sentence. Notice that in (1) the relevant string is early morning, where early is acting as an adjective modifier to the noun morning. However, in (2), the relevant string is early in the morning where early is acting as a temporal adverbial modifying the temporal phrase in the morning.

What does this mean, though, in more concrete terms? If, for instance, you consider early morning to be the time between 5 am and 7 am. Under the reading offered in (1), where you are to drink water in early morning, you are following this advice if you drink water between 5 am and 7 am. However, the reading in (2) makes no such restriction. As long as you drink water early sometime within the morning, say 8 am (if that is sufficiently early enough), you are following the advice given in the two sentences.