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There are plenty of obvious elementary properties separating primes of the form $4n+1$ and $4n+3$, e.g., quadratic residue's $$ (-1/p)=(-1)^{\frac{p-1}{2}}, $$ or the product $(4n+1)(4m+1)$ is closed, i.e., again of the form $4k+1$, but not for $4n+3$. There are also deeper properties, for example Chebyshev's bias. There are "more primes of the form $4n + 3$ than of the form $4n + 1$". This has become popular under the name Prime number races. This is related to GRH, the Grand Riemann Hypothesis. For a reference, see for example here.