What is the difference between topological and metric spaces? What is the difference between a topological and a metric space?
 A: A metric space gives rise to  a topological space on the same set (generated by the open balls in the metric). Different metrics can give the same topology. A topology that arises in this way is a metrizable topology. Using the topology we can define notions that are purely topological, like convergence, compactness, continuity, connectedness, dimension etc. Using the metric we can talk about other things that are more specific to metric spaces, like uniform continuity, uniform convergence and stuff like Hausdorff dimension, completeness etc, and other notions that do depend on the metric we choose. Different metrics that yield the same topology on a set can induce different notions of Cauchy sequences, e.g., so that the space is complete in one metric, but not in the other. In analysis e.g. one often is interested in both of these types of notions, while in topology only the purely topological notions are studied. In topology we can in fact characterize those topologies that are induced from metrics. Such topologies are quite special in the realm of all topological spaces. So in short: all metric spaces are also topological spaces, but certainly not vice versa.
A: An important difference in terms of technique is that in a metric space there are distinguished neighborhoods, namely the open balls of radius $r$ around a point $x$:
$$
B_r(x) = \{y\ :\ d(x,y)<r\}.
$$
These open balls form a local base for the topology and hence, carry all information about the topology of the metric space.
While in topological spaces the notion of a neighborhood is just an abstract concept which reflects somehow the properties a "neighborhood" should have, a metric space really have some notion of "nearness" and hence, the term neighborhood somehow reflects the intuition a bit more.
Moreover, in a metric space it is more convenient to work with sequences than in a topological space. For example it makes total sense, to memorize convergence of a sequence $(x_n)$ in a metric space to a point $x$ as "from some point on all $x_n$ are arbitrarily close to $x$". A statment which is quite useless in a topological space.
A: Just in terms of ideas: a metric space has a notion of distance, while a topological space only has a notion of closeness. If we have a notion of distance then we can say when things are close to each other. However, distance is not necessary to determine when things are close to each other.
A: A topological space is a set $X$ along with another set usually denoted by $\tau$ which is a collection of subsets of $X$, satisfying the following properties:


*

*$\emptyset, X \in \tau$

*Countable or Uncountable union of sets in $\tau$ is again in $\tau$

*Finite intersection of sets in $\tau$ is again in $\tau$


The space $(X,\tau)$ is called the topological space and the set $\tau$ is called a topology on $X$. The elements of $\tau$ are called open sets.
A metric space is a set $X$ and a function $d:X \times X \rightarrow \mathbb{R}^+ \cup \{0\}$ called the "metric" which takes in two elements from the set and pops out a non-negative real number. This metric has to satisfy certain properties:


*

*$d(x,y) \geq 0$, $\forall x,y \in X$

*$d(x,y) = 0$, iff $x=y$

*$d(x,y) = d(y,x)$, $\forall x,y \in X$

*$d(x,y) \leq d(x,z) + d(z,y)$, $\forall x,y,z \in X$


The space $(X,d)$ is called the metric space and $d$ is the metric i.e. a function such that $d:X \times X \rightarrow \mathbb{R}^+ \cup \{0\}$
Using this metric, we can define "certain" sets. The set of these sets call it $\tau$, along with the original set $X$ can now shown to be a topological space. So with every metric space $(X,d)$, we can associate a topological space $(X,\tau)$. The elements of the set $\tau$ are open sets.
However, topological spaces need not arise out of a metric space. There are non-metrizable topological spaces.
A: If metric space is interpreted generally enough, then there is no difference between topology and metric spaces theory (with continuous mappings). Building on ideas of Kopperman, Flagg proved in this article that with a suitable axiomatization, that of value quantales, every topological space is metrizable. This gives rise to a precise equivalence between the category of topological spaces and the category of generalized metric spaces, presented here (alg. univ.). 
