As you are observing, the arrangement of numbers on both are the same sort of spiral, with minor differences in where the start of the spiral is positioned, and direction. The “Square of Nine” you have shown illustrates what type of relationship diagonals and rows or columns of the spiral actually have. Each of the “rays” is a sequence generated by some second degree polynomial, which you can readily see by taking differences, and which makes perfect sense when you think about how the spiral is constructed. For example:

The Ulam spiral indicates that a lot of primes seem to be generated by certain second degree polynomials. Seem may be an important word here, as the polynomials in question are obviously not exclusive generators of prime numbers, and humans are notorious for seeing patterns in random data.

The pictured spiral is the same as the Ulam spiral, except that it spirals clockwise instead of counterclockwise, and he begins spiraling from the cell to the left of the 1 rather than the right. We should be able to come up with an expression for the number in the cell located at given coordinates but I don’t feel like thinking about it right now.

You’re not likely to find any deep relationships between the Ulam Spiral and the Gann Wheel because the former is actual mathematics, whereas the latter is pure numerology (i.e., bullshit).