Don’t think that much about the angles and vertical vs horizontal thing; actually all that is just a byproduct of the real difference between them: the stroke order.
Try writing them fast (or even, without taking the pen off the paper), and you will see that, naturally, the stroke order will lead to those shapes.
EDIT:
I can’t take a picture of my own handwritting right now; but found this image (here)
You can see, particularly for シ and ツ how the fast writing connecting the strokes lead to a given shape (slant, shape of the beginning and end of the big last stroke, etc)
EDIT2: I even learned here that the katakana and hiragana for those actually came from the same kanji! (「之」for「シ・し」and 「川」for「ツ・つ」); look at the pink background box at that page…