Schnorr signatures are proven to be secure and robust against chosen-message attacks in the random oracle model. In practice, this implies that as long as the condensation function behaves ideally,
the only way to "break" Schnorr's scheme is to solve the discrete logarithm problem.
Conversely, the construction of ECDSA does not guarantee its safety. The ECDSA scheme contains two independent condensation functions. One function
H is used to hash the message and the other function F to calculate the signature component
r. The value r is calculated by applying the condensation function
F to the x-abstract of the point
(x1, y1). The safety of the scheme is proved when both functions satisfy the random oracle model but practically, the function F does not.
While both are based on discrete logarithm problem, no one of them is secure against quantum computing. We are far away from such of attacks, so you don't have to worry.