The fixed points of a Flow (say, autonomous flow) is the set of all points 
$x=x^{\*}$ for $x \in$ state space $\mathbb{R}^{n}$ such that $F\left ( x=x^{\*} \right )$ =0.

The fixed points of a Map are the set of all points $x=x^{\*}$ such that 
$G\left ( x=x^{\*} \right )=x^{\*}$.
Really, G is just the identity map. 

In my lecture notes, the Maps equations are always denoted with a subscript. Is this a coincidence? 

The biggest question I do have for now is the physical differences illustrated between a Flow and a Map. 

Both the equation $\dot{x}=F\left ( x \right )=x\left ( N-x \right )$ and $\dot{x\_{t+1}}=F\left ( x\_{t} \right )=\lambda x\_{t}\left ( 1-x\_{t} \right )$ are used to model equation-the latter, being a Map, having been normalised. 
An intuitive explanation as to the "why" and the "when" Flows and Maps are used would be very helpful. What does either one accomplishes that the other doesn't?

Thanks in advance.