I tie my flies on with a standard clinch knot. The different eye orientations can have a noticeable effect, since I can slide the knot around, and have it affect the way the fly swims in the water.
With vertical eyes, sliding the knot "up" will cause the fly to dive more, sliding it down will cause it to rise to the surface. I actually use this property more on soft plastic bass lures than I do for flies, but it still works on flies.
With horizontal eyes, sliding the knot left or right make the fly swim the opposite direction. (knot to the left, lure moves to the right ...)
I've used this to make a lure swim up under docks, etc., when retrieving parallel to it. Again, this works very well with my bass gear, but still affects a fly the same way.
Conversely, if the knot is unintentionally "knocked" to those positions, you're fly will react accordingly. In these situations, the vertical eye usually just causes the fly to move up or down. The horizontal eye can produce more spinning, twisting up the tippet/leader.
These are MY observations. As texters are prone to say ... "Your mileage may vary". (YMMV)