What is the formula of manganese oxohydroxide: MnOOH and MnO(OH)2?
First, the difference the substances is the oxidation number of manganese. In $\ce{MnOOH}$, the oxidation number of manganese is 3+. While in $\ce{MnO(OH)_2}$, the oxidation number for manganese is 4+.

However this doesn't mean $\ce{MnOOH}$ is manganese (III) oxyhydroxide and $\ce{MnO(OH)_2}$ is manganese (IV) oxyhydroxide, not for $\ce{MnO(OH)_2}$ at least. If you move the element around, you will get $\ce{MnO(OH)_2}$ is actually $\ce{H_2MnO_3}$. $\ce{H_2MnO_3}$ is dihydroxy(oxo)manganese which is an acidic substance.

For the question regarding the structure for both substances, I'm sorry that I don't haven't the image but I can describe it (Hope that would do). For $\ce{MnOOH}$, manganese is single bonded with $\ce{OH}$ and manganese is double bonded with oxygen. For $\ce{MnO(OH)_2}$, manganese have two single bonds with both of the $\ce{OH}$ and manganese is double bonded with oxygen.

In conclusion, $\ce{MnOOH}$ is manganese (III) oxyhydroxide and $\ce{MnO(OH)_2}$ is dihydroxy(oxo)manganese. In structure wise, manganese is single bonded with $\ce{OH}$ and manganese is double bonded with oxygen for $\ce{MnOOH}$. For $\ce{MnO(OH)_2}$, it likes $\ce{MnOOH}$ but have one more single bond with $\ce{OH}$.