Difference between State Space and Sample Space Aren't they both the same?
According to me they are just different terminologies used for different types of systems. What I understand is that sample space is used when we talk about static systems and state space when we talk about dynamic systems. Am I correct? If not then what exactly is the difference between them?
My teacher had asked for the state space in Markov Chain questions, so I'm guessing that state space is used only in dynamic systems? She said that if we roll 2 dice then the sample space will be {1,2,...,6},
but the state space will be {(1,1),(1,2),...,(6,6)}.
But as far as I remember, the sample space for rolling 2 dice is
{(1,1),(1,2),...,(6,6)}.

The sample space is the set that contains all possible outcomes (events) for an experiment, and is usually denoted by $\Omega$ (the set of all possible events $\omega$).
For example, when tossing two coins, $\Omega=\{\text{(heads,heads)},\text{(heads,tails)},\text{(tails,heads)},\text{(tails,tails)}\}$

As pointed out by user275313, this answer was not completely right. The state space is the set of all possible states in which a dynamic system can be, and it denotes that the system evolves continuously from one to the next. 
On the other hand, the state space refers to the set of all possible realizations of a random variable. Remember that a random variable is a function that maps events (abstract) to mathematically representable entities (e.g. numbers).
For example, let $X$ be a random variable associated with tossing a coin, so that:
$X(\omega )={\begin{cases}1,&{\text{if}}\ \ \omega ={\text{heads}},\\0,&{\text{if}}\ \ \omega ={\text{tails}}.\end{cases}}
$
Therefore, if $Y$ is a multivariate random variable that for an experiment of tossing two coins, $Y=\{X_1,X_2\}$, the state space will be $Y=\{(1,1),(1,0),(0,1),(0,0)\}$.

