TITLE: Is there difference between finitely presented groups and finitely generated groups?
QUESTION [6 upvotes]: A group is said to be finitely generated if it can be generated by a finite set of generators. 
Question : Is there difference between finitely presented groups and finitely generated groups?

REPLY [3 votes]: Rips' construction gives lots of examples of finitely generated groups which are not finitely presentable. Rips* proved the following result.
Theorem. For every finitely presented group $Q$ there exists a hyperbolic group $H$ and a finitely generated, normal subgroup $N$ of $H$ such that $H/N\cong Q$.
Given a finite presentation of $Q$, Rips explicitly constructs the group $H$. This result is usually referred to as Rips' construction. It turns out that in Rips' construction the subgroup $N$ is finitely presentable if and only if the image group $Q$ is finite (see Exercise II.5.47, p227, of Bridson and Haefliger, Metic spaces of non-positive curvature - one direction is obvious, while the other direction is highly non-trivial).
*
E. Rips, Subgroups of small Cancellation Groups, Bulletin of the London Mathematical Society, Volume 14, Issue 1, 1 January 1982, pp45–47, doi link.