Brad DeLong has a thoughtful post on the relationship of modern liberalism to libertarianism and, therefore, to classical liberalism.
One important thing that I think is left out of his account has to do with the environment. Issues such as air pollutions and ocean acidification weren’t major topics of concern during the high tide of classical liberalism. Not that pollution didn’t exist, but on the one hand the issues weren’t as well-understood as they are today, and on the other hand the poorer societies of the nineteenth century didn’t care that much about them. At the same time, the basic conceptual apparatus of classical liberalism provides clear reasons to think that negative externalities ought to be regulated or taxed. Modern liberalism strongly embraces that idea, and the general concept that government intervention is necessary to prevent the operation of the economy from producing socially undesirable levels of pollution and death. Contemporary libertarianism is basically opposed to such interventions, for a variety of more-or-less ad hoc reasons plus, of course, reasons of money and (in the U.S. context at least) a prolonged period of coalition alongside atavistic conservatives.