What are the differences between these sentences:
Les vacances d'été viennent juste de commencer
Les vacances d'été viennent de commencer
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They are variants of the same meaning. They both mean "Holidays have just started", but the former insists even more of the small amount of time elapsed since it started.
They are saying the same thing basically, when taken out of context, with an added emphasis in the first case on the temporality.
But I'd also add that the first form is often found as a premise to a negative or opposite part like saying 'although I'm already feeling better' / 'pourtant je me sens déjà mieux'.
The word juste has several meanings, too many to list here. One of the meanings (I.2, II.A.2, II.A.1.c) is precision, and from there (I.B.c.β, II.A.2, II.A.3, II.B.2), an indication of tightness, or that a gap is small.
Venir de is a near past construction: “les vacances d'été viennent de commencer” = “the summer holidays have just begun”. As you can see, the word “just” in English can indicate a small gap, too. In French, you can add the word “juste” to this sentence, and it acts as an intensifier for the nearness of the past. You can even add another intensifier: “les vacances d'été viennent tout juste de commencer”; here the word tout reinforces juste to convey that the holidays have just barely begun.