Let $f,g$ be non constant entire functions. If $h$ is holomorphic on the image of $f$, and $g=h\circ f$, we say $f$ is a right factor of $g$.
Suppose $f$ is a right factor of $g$, and $g$ is a right factor of $f$, i.e. there is $h_1$ such that $$g=h_1\circ f,$$ and there's $h_2$ such that $$f=h_2 \circ g.$$ What can be said of $f$ and $g$ ? how are they related ?
Lemma 1: Every element of $\text{Aut}(\mathbb{C}^\times)$ is of the form $z\mapsto az$ or $\displaystyle z\mapsto \frac{a}{z}$ for some $a\in\mathbb{C}^\times$.
(see here, Theorem 4.10)
Lemma 2: Every element of $\text{Aut}(\mathbb{C})$ is of the form $z\mapsto az+b$ for some $a,b\in\mathbb{C}$, $a\ne 0$.
(see here)
Let $f(\mathbb{C})=X$ and $g(\mathbb{C})=Y$, and note by the open mapping theorem that $X$ and $Y$ are open. Then, $h_1:X\to Y$ is holomorphic, and $h_2:Y\to X$ is holomorphic. Note that $(h_1\circ h_2)\circ g=g$ and so $h_1\circ h_2$ is the identity map on $Y$. Similarly, $h_2\circ h_1$ is the identity map on $X$. Thus, $h_1$ is a biholomorphism $X\to Y$ and $h_2$ is a biholomorphism $Y\to X$.
Now, since $f$ is entire and non-constant, the Little Picard Theorem tells us that $X$ is either $\mathbb{C}$ or $\mathbb{C}-\{p_1\}$ for some $p_1\in\mathbb{C}$, and similarly $Y$ is either $\mathbb{C}$ or $\mathbb{C}-\{p_2\}$. Since $X\approx Y$ we must have that $X=\mathbb{C}$ if and only if $Y=\mathbb{C}$ and similarly $X=\mathbb{C}-\{p_1\}$ if and only if $Y=\mathbb{C}-\{p_2\}$. So, from here we break into cases:
Case 1: ($X=Y=\mathbb{C}$) From this we see that $h_1$ is an element of $\text{Aut}(\mathbb{C})$, and thus by Lemma 1,is of the form $h_1(z)=az+b$ for some $a,b\in\mathbb{C}$, $a\ne0$. So that $g=af+b$.
Case 2:($X=\mathbb{C}-\{p_1\}$,$Y=\mathbb{C}-\{p_2\}$) We have that $h_1$ is a biholomorphism $\mathbb{C}-\{p_1\}\to\mathbb{C}-\{p_2\}$. Let $T_c$ be translation by $c$, and consider $T_{-p_2}\circ h_1\circ T_{p_1}$, which is an element of $\text{Aut}(\mathbb{C}^\times)$, and so either equal to $z\mapsto az$ or $\displaystyle z\mapsto \frac{a}{z}$ for some $a\in\mathbb{C}^\times$. In the first case we get that $h_1=a(z-p_1)+p_2$ and in the second case $\displaystyle h_1(z)=\frac{a}{z-p_1}+p_2$. Thus, either $\displaystyle g=a(f-p_1)+p_2$ or $\displaystyle g=\frac{a}{f-p_1}+p_2$.
EDIT: It's interesting to note that if we forget the more specific case-wise analysis done above then we can phrase the above as saying "if $f$ is a right-factor of $g$ and $g$ is a right factor of $f$ then there exists some $A\in\text{PGL}_2(\mathbb{C})$ such that $Af=g$". In particular, if two entire functions are both right factor related, then they differ by an automorphism of $\mathbb{P}^1$.
