11/29/2020. I will leave the original post unchanged below, but some of our claims here proved to me untrue, or at least more nuanced than we realized. Were I to write this today, I’d point out that seams don’t matter most of the time when the ball has no gyro spin. With gyro spin, they can matter a great deal.
This is our movie explaining that 2-seam and 4-seam pitches on the same axis will produce the same trajectory. The movie was an entry to the 2018 APS DFD Gallery of Fluid Motion and was intended for a scientific audience. Counter to some of the stories on the web, we concluded that, at least for these Wilson 1030 baseballs, the seams have a profound effect on the boundary layer separation yet their orientation does not have much effect on the overall aerodynamics. A 2-seam is often thrown with more pressure on the index finger causing it to leave the ball after the middle finger. The brief period with one finger in the ball induces some side spin resulting in run to the arm side. Some translations for non-fluid dynamicists: Fluid: any gas or liquid. Anything that flows. PIV: Particle Image Velocimetry is a laboratory air velocity measurement technique. See our post on this. Vorticity: is the curl of velocity (for any mathematicians), or, for the rest of us, how much the air is spinning. While not important to baseball aerodynamics in general, we use it to mark the boundary layer, or the part of the air that feels the skin of the baseball. Boundary layers contain vorticity, with the boundary layer on top of the ball spinning clockwise (which we normally color blue) and counter clockwise below (which we color red). We care about this because at some point near the center of the ball, the boundary layer separates from the ball and forms a wake, and plotting vorticity makes it very easy to see the wake, as show below. This ball is moving left to right at 90mph and is spinning 1650 rpm. All of these data sets were acquired in a vertical plane.
This makes it easy to compare two cases and determine which one has a larger drag. In general, wider and longer wakes produce more drag that short and narrow ones. In the cases below, ball 1 has a lower drag than ball 2. 
BUT, actually, these are the same pitch at different points in the rotation of the baseball. Both the lift and drag on the ball vary as it rotates. Let’s have a look at that. In the movie below, we show a 2-seam and a 4-seam fastball. The axis of both balls is straight into the page. For one pitcher, that may be parallel to the ground while for another (with a more side-arm motion) it may be tilted toward the arm side.