How is prisoner's dilemma different from chicken? Chicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash. So the payoff matrix looks something like this:
                  B swerves               B straight
A swerves            tie               A loses, B wins
A straight     B loses, A wins            both lose 

But I have heard of another situation called the prisoner's dilemma, where two prisoners are each given the choice to testify against the other, or remain silent. The payoff matrix for prisoner's dilemma also looks like  
                  B silent               B testify
A silent            tie               A loses, B wins
A testify     B loses, A wins            both lose 

I remember hearing that in the prisoner's dilemma, it was always best for both prisoners to testify. But that makes no sense if you try to apply it to chicken: both drivers would crash every time, and in real life, almost always someone ends up swerving. What's the difference between the two situations?
 A: The Prisoner's Dilemma was a game constructed for a very specific purpose: 

Each player has a preferred strategy that collectively results in an inferior outcome.

In game theory language, both players have a dominating strategy: regardless of the opponent's action, they should choose a specific action (in this case, an action typically called Defect). If both players choose their dominating strategy, it leads to a (Nash) equilibrium, from which no individual player benefits from deviating. This equilibrium is (Pareto) inefficient in the sense that all players prefer an alternative outcome. Let's see this in practice:
\begin{vmatrix}
\ & \color{green}{Cooperate} & \color{green}{Defect} \\
\ \color{blue}{Cooperate} &  \color{blue}{4},\space\space\color{green}{4} & \color{blue}{5},\space\space\color{green}{1} \\
\ \color{blue}{Defect} &  \color{blue}{1},\space\space\color{green}{5} & \color{blue}{2},\space\space\color{green}{2}
\end{vmatrix}
We see that the $\color{blue}{blue}$ player's payoff is always higher for Defect than for Cooperate; that's what it means to be a dominating strategy. The same is true for $\color{green}{green}$. If both players choose Defect, the outcome is $2,2$ which is inferior for all players to the outcome $4,4$.
Chicken was constructed in the same vein for a different purpose:

No player has a preferred strategy, and all players are in direct
  rivalry with one another.

Unlike the Prisoner's Dilemma, there are no dominating strategies, and this makes a big difference. For example, what would you choose in the following game:
\begin{vmatrix}
\ & \color{green}{Cooperate} & \color{green}{Defect} \\
\ \color{blue}{Cooperate} &  \space\space\space\color{blue}{0},\space\space\color{green}{0} & \space\space\space\color{blue}{2},\color{green}{-1} \\
\ \color{blue}{Defect} &  \color{blue}{-1},\space\space\color{green}{2} & \color{blue}{-5},\color{green}{-5}
\end{vmatrix}
Your best strategy is to anti-coordinate with your opponent; that is, to Defect when they Cooperate and Cooperate when they Defect. But if you had a choice, you would prefer to be the one Defecting. Mutual Defection is the worst outcome and isn't an equilibrium, but neither is Mutual Cooperation. In fact, the equilibria are when you and your partner anti-coordinate and is inherently adversarial.
So in general, we have a symmetric payoff matrix:
\begin{vmatrix}
\ & \color{green}{Cooperate} & \color{green}{Defect} \\
\ \color{blue}{Cooperate} &  Reward & \color{blue}{T},\space\space\color{green}{S} \\
\ \color{blue}{Defect} &  \color{blue}{S},\space\space\color{green}{T} & Punish
\end{vmatrix}


*

*In PD, $Temptation (T) > Reward (R) > Punish (P) >
   Sucker (S)$

*In Ch, $Temptation (T) > Reward (R) > Sucker (S) > Punish (P)$


While it is true that the Prisoner’s Dilemma and Chicken have a different preferential ordering of outcomes and thus different equilibria, the purposes of the two games are completely different.
Interesting Questions You Could Ask (on a Game Theory StackExchange):


*

*What happens when each game is iterated (i.e. multiple rounds)?

*What happens as the number of players increases (i.e public goods
game)?

A: (See http://en.wikipedia.org/wiki/Chicken_%28game%29#Prisoner.27s_dilemma.)
The difference is in the payoff. In the "chicken" game, the payoff matrix is like
        Sw       St
   Sw  0,  0   -1, +1
   St +1, -1  -10, -10

While in the PD game: 
        Si       Te
   Si  3, 3     0, 5
   Te  5, 0     1, 1

Both games have this structure in the payoff table:
         C       D
   C   R, R     S, T
   D   T, S     P, P

But:


*

*In PD game, the order is T > R > P > S

*In Chicken game, the order is T > R > S > P


This leads a different Nash equilibria. In the PD game, if A remains silent, B chooses to testify because T > R, while if A testifies, B should also testify because P > S. So testifying is B's most rational choice after considering all possibilities.
But in the Chicken game, as S > P, if A goes straight, B should swerve. This leads to two Nash equilibria in the pure game: (St, Sw) and (Sw, St). 
A: The games aren't just about winning or losing, but also about utility. Here is a more accurate table for chicken:
                  B swerves                B straight
A swerves    No gain for either          A loses, B wins
A straight     B loses, A wins         Both have large loss

Here is one for prisoners dilemma:
                    B silent                        B testify
A silent       Both have small loss         A large loss, B loses nothing
A testify  B loses nothing, A large loss,        both medium lose 

In the prisoners dilemma, an individual prisoner will always do better by testifying (look at the table), however, by both testifying they end up in a worse position than if both were silent. In contrast, in chicken, going straight will be better if the other swerves and swerving will be better if the other goes straight. Your tables represent some strategies as being equally good for players when they are not.
