Is there any importance to notations like $| Omega Vrangle$, $|aVrangle$?

It will be better if you think that $|rangle in mathcal{V}$ and $langle |in mathcal{V}^*$ where $mathcal{V}^*$ is dual space of $mathcal{V}$. Similarly any scalar $alpha in mathcal{F}$ and $alphain mathcal{F}^*$. Also Operators $Omega$ act on $|rangle in mathcal{V}$ and for all $Omega $ there exist $Omega^dagger$ that act $langle |in mathcal{V}^*$.

Now In $mathcal{V}$, $$Omega |Vrangle =|Omega Vrangle $$ $$alpha|Vrangle=|alpha Vrangle $$ and In dual, $$langle V|alpha^*=langle alpha V|$$ $$langle Omega V|=langle V|Omega^dagger$$

Further, When one writes $Omega |Vrangle$, We mean that an operator acting on the vector while when we write $|Omega Vrangle$, It's a transformed vector. It can't be said using this has many merits. But consider an example

Let's prove that the energy of the Harmonics oscillator can't be less than zero.

$$langle psi|H|psirangle =langle psi|P^2|psirangle +langle psi|X^2|psirangle =langle Ppsi|Ppsirangle +langle Xpsi|Xpsiranglegeq 0$$ Using the fact, that norm of a vector is greater or equal to zero. It was useful to write that way to see the norm of vector.