Not particularly. It has some impact, sure, within the context of the Trek
universe, but kinds of stories? Exactly what "kinds of stories" can only be told before the 25th century as opposed to during it? Are there types of characters that can only exist in a 25th century Starfleet? I suppose there are species we wouldn't see, but honestly how important is that detail? How vital can it be that (for example) a Jem Ha'dar be Chief of Security? Concepts? What concepts in particular? Which ones are so unique to the 25th century that no story told before then can deal with such? War? Uh...no. The dichotomy between reason and emotion? Prejudice versus tolerance? I can see how details
would change, but nothing terribly vital.
A starship explores some uncharted area of the galaxy.
Starfleet takes over a space station in orbit around a planet as it enters the process of joining the Federation.
An accident hurls a starship to a distant part of the galaxy.
Several junior officers begin their careers as friends aboard a brand new starship.
Near a dangerous region of space, a Starfleet medical ship struggles to keep up with the sick and wounded.
Almost any of these basic premises could fit easily into the 23rd, the 24th, the 25th or the 39th centuries.
To be sure, the era of any such show will impact its stories and characters. But in and of itself
I don't find the question of era all that compelling.