What is the difference between topological and metric spaces? What is the difference between a topological and a metric space?

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.
