The Weierstrass function has two zeros per each parallelogram of periods, which may coincide, producing a double zero.
This cover has degree 2. We need to quotient the elliptic curve $\Bbb C/\Lambda$ by the transformation $z\to -z$, which has order 2.
The branched cover (defined by the Weierstrass function) has degree 2. To obtain $\Bbb C\Bbb P^1$, we need to quotient the elliptic curve $\Bbb C/\Lambda$ by the transformation $z\to -z$, which has order 2.