Yes the date does matter. However, in mitigation for her, if she got the time right (same time but day before) then the Ascendant will be right, unless you are at the very start or very end of the Ascending sign.
Sun, Venus and Mercury move at about 1 degree a day, so they will all be about one degree out on their correct positions. Mars will be about half a degree out and the rest are only going to be a minute or two out. Such differences in the inner planets may effect some of the aspects - that is ones that she has taken into account may be no longer in aspect on your birthday and ones that she has ignored may have come into aspect a day later. The chances of major differences are low but they should be rechecked.
However the Moon will be considerably out - about 12 degrees or so and this will be the key difference. It will also make a significant difference to the chart reading of your character.
I think her excuse was more an attempt to brush over a mistake, if it had been me, I'd have apologised and redone it.
I'd ask her nicely to do another reading based on the correct information.