NOTE: I tried hard and came up with a lose proof, I have posted it as a answer. Do comment/correct if you can.
Let $P=\{x|Ax\geq b\}, A\in \mathbb{R}^{m\times n}$ $Q=\{y|Gy\geq h\},G\in \mathbb{R}^{k\times n}$ $x,y\in\mathbb{R}^n$
I have to prove that if the rows of A span $\mathbb{R}^n$ and the rows of G don't span $\mathbb{R}^n$ (clearly, they span some space less than n) then $P\neq Q$.
So, what I have tried:
I first tried to analyze what it means for a matrix whose rows don't span Rn. I figured that G must have a column full of zeros if its rows cannot span $\mathbb{R}^n$. Let that column be j. ($\therefore G_j=\mathbb{0}$)
Now, since P and Q are basically defined by linear constrained and P is more strict with respect to its constraints on x, $P\neq Q$ will happen only if I can find a $p$ such that $p\in Q, p \notin P$.
So,
PROPOSITION: $\exists p\in\mathbb{R}^n$ $Gp\geq h$ $Ap Now, I am stuck.