It is just the regular countable/uncountable difference with and without the article, but I would say that the writer took a liberty using uncountable "promise" that way, and it strikes my ear as overreaching by a writer with less skill than inspiration. Uncountable "promise" in its usual sense would mean that the day showed promise, that it was a promising day, that one would expect good things to come simply because of the nature of that particular day, but that is not what the writer means—he means that any day has a quality he calls "promise". Try replacing "promise" with "hope" or some other word of the kind that you are more familiar with.
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