For $X_1:=X$, $X_2:=X'$, some positive real $c_1,c_2,a_1,a_2$, and all positive real $t$ we have $$P(|X_j|>t)\le c_j e^{-a_j t^2}$$$$P(|X_j|>t)\le c_j e^{-a_j t^2} \tag{1}$$ for $j=1,2$. So, for $t:=\epsilon>0$, $$P(|X_1-X_2|>t)\le P(|X_1|>t/2)+P(|X_2|>t/2)\le c_1 e^{-a_1 t^2/4}+c_2 e^{-a_2 t^2/4}.$$$$P(|X_1-X_2|>t)\le P(|X_1|>t/2)+P(|X_2|>t/2) \\ \le c_1 e^{-a_1 t^2/4}+c_2 e^{-a_2 t^2/4}.$$
The OP commented that the notion of a sub-Gaussian distribution was meant not in the usual sense. Namely, according to that comment, condition (1) should be replaced by $$P(|Y_j|>t)\le c_j e^{-a_j t^2}, \tag{2}$$ where $$Y_j:=X_j-m_j$$ for some real $m_j$. Then, for $t:=\epsilon>|m_1-m_2|$, $$P(|X_1-X_2|>t)\le P(|Y_1-Y_2|>t-|m_1-m_2|) \\ \le P(|Y_1|>(t-|m_1-m_2|)/2)+P(|Y_2|>(t-|m_1-m_2|)/2) \\ \le c_1 e^{-a_1(t-|m_1-m_2|)^2/4}+c_2 e^{-a_2(t-|m_1-m_2|)^2/4}.$$