To solve this problem, we need to find the conditional probability \(P(X > 0 \mid X \neq 2)\).

First, calculate the total number of outcomes when two spinners are spun. Each spinner has 5 sides, so there are \(5 \times 5 = 25\) possible outcomes.

Next, determine the number of outcomes where the score is not equal to 2. The score is 2 when the numbers are (1,3), (3,1), (2,4), (4,2), (3,5), and (5,3). There are 6 such outcomes.

Thus, the number of outcomes where the score is not 2 is \(25 - 6 = 19\).

Now, find the number of outcomes where the score is greater than 0 and not equal to 2. This includes all outcomes except those where the numbers are equal (score = 0) and those where the score is 2. There are 5 outcomes where the numbers are equal: (1,1), (2,2), (3,3), (4,4), (5,5).

Therefore, the number of outcomes where the score is greater than 0 and not equal to 2 is \(25 - 5 - 6 = 14\).

The conditional probability is then given by:

\(P(X > 0 \mid X \neq 2) = \frac{\text{Number of outcomes } (X > 0 \cap X \neq 2)}{\text{Number of outcomes } X \neq 2} = \frac{14}{19}\)

Thus, the probability is \(\frac{14}{19}\) or approximately 0.737.