Here is a quote from John Lee's Introduction to Topological Manifolds, Second Edition:
The definition of topological spaces is wonderfully flexible, and can be used to describe a rich assortment of concepts of "space". However, without further qualification, arbitrary topological spaces are far too general for most purposes, because they include some spaces whose behavior contradicts many of our basic spatial intuitions.
By requiring a space to be Hausdorff we exclude "pathological" topological space, such as the one in which a sequence could converge to more than one point.
Then, why do we not require all topological spaces to have the Hausdorff property? What do we gain by defining topological spaces to be more general spaces than Hausdorff spaces?