On a complex projective variety, Gromov-Witten invariants can be interpreted as virtual counts of curves, so they are biregular invariants.
However, they are not birational invariant in general. The behaviour of Gromov-Witten invariants under an arbitrary birational modification is in fact rather subtle.
For more details and examples you can have a look at Section 1.4 of the paper
D. Abramovich, J. Wise Birational invariance in logarithmic Gromov-Witten theory, Compos. Math. 154, No. 3, 595-620 (2018). ZBL1420.14124.