Pivotal Web Services - Run.pivotal.io - is the CF equivalent to Heroku. Documentation is at docs.run.pivotal.io. IBM has one at Bluemix.net as well.
CF is very good if you want to run a cloud native / 12 factor app. It's also rapidly evolving to run rich workloads (TCP router is coming, .NET is in beta).
I would say CF is more flexible at the kinds of apps and services it can run as the buildpacks tend to be more sophisticated forks of the Heroku buildpacks, or new from scratch buildpacks (like PHP, Or Java). Docker images should be hostable in the near future as well.
Whereas the service levels and instance sizing flexibility available from Heroku tends to be richer.
This is a function of where both companies make their money. Heroku only makes money from their public cloud, whereas Pivotal is focused on selling high end subscriptions for those who want to run their own private CF on Amazon or VMware or OpenStack.
I've looked quickly at it, but unlike (say) Heroku, found it hard to know where to start to host a simple app.