Note that you do not have an efficiently computable homomorphism from $G_1$ to $G_2$, but in Type-2 you have an efficiently computable homomorphism $\psi: G_2 \rightarrow G_1$ and in Type-3 you do not have one.

But what I don't understand is what is the use of the homomorphism in cryptography?

Well, if you have a tuple $(aP',bP',cP')\in G_2^3$ with $P'$ being a generator of $G_2$, you can check if $e(\psi(aP'),bP')=e(\psi(P'),cP')$ holds. Consequently, the decisional Diffie Hellman (DDH) problem is easy in $G_2$, but remains hard in $G_1$. In cryptography, this is typically formalized as the so called external Diffie Hellman (XDH) assumption.

In a Type-3 setting, as you can not map between $G_2$ and $G_1$, the DDH seems to be hard in $G_1$ and in $G_2$. In cryptography, this is typically formalized as the so called symmetric XDH (SXDH) assumption.

So if you have a Type-2 pairing, then you have the XDH setting and if you have a Type-3 pairing, you have the SXDH setting.