There is an irrational number of days in a year. It's one of those numbers like the square root of 2, or pi, that cannot be written down. So, every calendar is based on an estimate.
The Jewish calendar uses a pretty precise one, which we call Tequfas Rav Ada (the seasons of Rav Ada) -- 365 days, 5 hours, 997 chalaqim, and 48 rega'im. (A cheileq is 1/18 of a minute, i.e. 1080 per hour, and a rega is 1/64th of a cheileq.)
Which allowed Sanhedrin to make a calendar, but for something like davening, it has to be within the reach of everyone. And really, do we need to-the-day precision for when the rainy season starts? It's not like seasons actually change on the moment of equinox?
So for the purposes of the siddur, they decided to go with Shemu'el's simpler, if less accurate, estimate -- a solar year is around 365-1/4 days. In Israel, it was particularly easy -- the Romans were using the Julian calendar, based the same estimate! So there would be a simple rule for when to start, based on the secular date. But even in Babylonia, in the Sassanid Empire, the calculation was more manageable.
Now that most of the world uses the Gregorian calendar, that simple rule got a little more complicated. But since the change is only once a century, three centuries out of four, life is still pretty easy.
Every year, the calendar is a quarter of a day off, so the Julian calendar adds a leap day when that error comes to getting the date wrong. But, they start days at midnight. We start days around six hours earlier (a quarter of a 24-hour day), at sunset. So, the way we count days, the error adds up to the next day a quarter of a cycle earlier. So, the rule for "Vesein Tal" is that it's a day later the fall before a leap day.
But again, we aren't really basing ourselves on the Julian calendar. It's just a way to make the rule easy. We are basing ourselves on Tequfas Shemu'el, which just happens to be the same 366-1/4 day estimate for the year as the Julian calendar is based upon. Balancing ease of use and the kind of accuracy the tefillah requires to be useful.