Second pre-image resistance vs Collision resistance From Wikipedia:


  Second pre-image resistance
  
  Given an input $m_1$ it should be difficult to find another input $m_2$ such
  that $m_1$ ≠ $m_2$ and $\operatorname{hash}(m_1) = \operatorname{hash}(m_2)$. Functions that lack this
  property are vulnerable to second-preimage attacks.
  
  Collision resistance
  
  It should be difficult to find two different messages $m_1$ and $m_2$ such
  that $\operatorname{hash}(m_1) = \operatorname{hash}(m_2)$. Such a pair is called a cryptographic hash
  collision.


Could someone explain the difference between these two please? They very much appear to be identical to me, as in both definitions $m_1 \neq m_2$ and $\operatorname{hash}(m_1) = \operatorname{hash}(m_2)$.
The difference is in the choice of $m_1$.


In the first case (second preimage resistance), the attacker is handed a fixed $m_1$ to which he has to find a different $m_2$ with equal hash. In particular, he can't choose $m_1$.
In the second case (collision resistance), the attacker can freely choose both messages $m_1$ and $m_2$, with the only requirement that they are different (and hash to the same value).


(From this, it is also obvious that collision resistance implies second preimage resistance: An attacker can just choose an arbitrary $m_1$ and compute a second preimage $m_2$ to obtain a collision.)