Another toughly… The difference can sometimes be very subtle but here's my take on it.
「はじめ」は単純に期間の最初の部分を意味する(a time from which something begins/the first part of a period of time)のに対して、
「はじまり」は何か大きな一連のイベントの最初のきっかけ(an event that prompts a chain of events)、または人生や歴史上などのイベントや物語の最初の部分(the first part of a significant/historical event or a story)を指すことが多いと思います。英語ならoriginやopeningと訳せる場合もあると思います。
Going back to your example, the person in question has been whiling away his vacation and the 夏休みのはじめから part is there to specify since when, which means the speaker obviously doesn't have an exciting story to tell about this summer vacation. To generalize a bit, 夏休み is usually seen as just a period of time that will start automatically every year without some first cause setting it in motion as long as you are a student or an employee at your organization. For this reason, it collocates better with はじめ just like "今週のはじめ" and "今年のはじめ".
To be sure, expressions like 楽しい夏休みの始まり are also possible and, in fact, not so uncommon, but they give you the impression that the speaker thinks of summer vacation not as a span of time, but as a whole series of important events (she might have gone on a month-long trip to some nice place and have a story to tell).
事の始まりは、３年前日本に旅行した時のことです。=It all started 3 years ago when I took a trip to Japan.
宇宙の始まり=the origins of the universe
そもそもけんかの始まりは何なの？=So what started the fight in the first place?
あの小説は始まりのシーンがとてもいい。=I really like the opening of that novel.