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

<cite authors="Abramovich, Dan; Wise, Jonathan">D. Abramovich, J. Wise [*Birational invariance in logarithmic Gromov-Witten theory*](http://dx.doi.org/10.1112/S0010437X17007667), Compos. Math. **154**, No. 3, 595-620 (2018). [ZBL1420.14124](https://zbmath.org/?q=an:1420.14124).</cite>