Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$.
1. **Define the problem and constraints:**
   We need to find the maximum number of elements in a subset \( A \) of \(\{1, 2, \ldots, 2020\}\) such that the difference between any two distinct elements in \( A \) is not a prime number.

2. **Construct a potential solution:**
   Consider the set \( A \) formed by taking every fourth number starting from 1:
   \[
   A = \{1, 5, 9, 13, \ldots, 2017\}
   \]
   This set can be written as:
   \[
   A = \{4k + 1 \mid k = 0, 1, 2, \ldots, 504\}
   \]
   The number of elements in this set is:
   \[
   |A| = 505
   \]

3. **Verify the construction:**
   For any two elements \( 4a + 1 \) and \( 4b + 1 \) in \( A \), the difference is:
   \[
   (4a + 1) - (4b + 1) = 4(a - b)
   \]
   Since \( a \) and \( b \) are integers, \( 4(a - b) \) is always a multiple of 4. The only prime number that is a multiple of 4 is 2, but 2 is not a multiple of 4. Therefore, the difference between any two elements in \( A \) is not a prime number.

4. **Consider the possibility of a larger set:**
   Assume \( |A| > 505 \). We will use the pigeonhole principle to show a contradiction. Divide the set \(\{1, 2, \ldots, 2020\}\) into 505 subsets of the form:
   \[
   \{4k + 1, 4k + 2, 4k + 3, 4k + 4\} \quad \text{for} \quad k = 0, 1, 2, \ldots, 504
   \]
   Since \( |A| > 505 \), by the pigeonhole principle, at least one of these subsets must contain at least two elements from \( A \). Without loss of generality, assume these elements are \( 4m + a \) and \( 4m + b \) where \( 1 \leq a < b \leq 4 \). The possible differences are:
   \[
   |(4m + a) - (4m + b)| = |a - b|
   \]
   The values of \( |a - b| \) can be 1, 2, or 3, all of which are prime numbers. This contradicts the condition that the difference between any two elements in \( A \) is not prime.

5. **Conclusion:**
   Therefore, the maximum number of elements in \( A \) is 505.

The final answer is \(\boxed{505}\)