No, it does not matter. Both gauge and absolute pressures give identical results in Bernoulli's equation.
Bernoulli's equation relates pressure, velocity, and height along a streamline:
$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$
Between two points (1 and 2):
$$P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$$
Gauge pressure = Absolute pressure - Atmospheric pressure:
$$P_g = P_a - P_\text{atm}$$ $$\text{or} \quad P_a = P_g + P_\text{atm}$$
Substitute absolute pressures into Bernoulli's equation:
$$(P_{g1} + P_\text{atm}) + \frac{1}{2}\rho v_1^2 + \rho gh_1 = (P_{g2} + P_\text{atm}) + \frac{1}{2}\rho v_2^2 + \rho gh_2$$
Atmospheric pressure \(P_\text{atm}\) cancels out:
$$P_{g1} + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_{g2} + \frac{1}{2}\rho v_2^2 + \rho gh_2$$
The equation looks exactly the same whether using gauge or absolute pressures. Atmospheric pressure terms cancel when comparing points exposed to the same atmospheric pressure.
No difference exists. Gauge pressures are actually more convenient for engineering applications involving open-to-atmosphere flows.