Two things go wrong when a cigar is smoked too early, and both are visible from the first few puffs.
The first is mechanical. Moisture has not yet equalised across the wrapper, binder, and filler, which means each component holds a different percentage of relative humidity. The wrapper rehydrates first when a cigar arrives in your humidor, the binder follows, the filler last. If you light a cigar before the gradient has flattened, the wrapper races ahead of the bunch underneath: the burn line wanders, the ash flakes off in pieces rather than holding in long sections, and the draw feels tight at the head and slack at the foot. None of this is the cigar’s fault. The cigar has not yet finished arriving.
The second is chemical. Freshly rolled cigars carry residual ammonia from fermentation, which has not yet finished off-gassing. Smoke a cigar deep in this period and the flavour reads sharp on the nose, sour on the palate, and muted in the middle. You may, depending on your sensitivity, feel slightly off afterwards. There is nothing wrong with the leaf; there is too much ammonia in the air around your palate.
The full case for waiting and the chemistry behind it lives in How to Rest a Cigar.