Both are known as cloud platforms, or PaaS. That is, they don't just give you the rudiments of a cloud, called Infrastructure As A Service, called IaaS, but also the operating systems, languages, databases and web services developers need to do build cloud services. (Software as a Service is abbreviated as SaaS.)
While VMWare is known as a proprietary company, because its vSphere operating system is closed source and company-controlled, that company chose to make Cloud Foundry open source, even placing it within the Apache Foundation for that purpose. (This is in contrast with the third major PaaS offering, Microsoft (MSFT) Azure, which like vSphere is proprietary.)
Red Hat was taken back a bit by the audaciousness of VMWare's move. Red Hat was built on open source - its main product is Red Hat Enterprise Linux - so to have a proprietary company try and be "more open source than thou" looked like chutzpah. (Not that there's anything wrong with chutzpah.) It took time to formulate a response.
This week, that response is emerging:
A collection of pricing models, not just one.
A development emphasis on solving big cloud problems like security, compliance and privacy, not just developer problems like language support.
A new emphasis on alliances, starting with Dell (DELL).
Dell has been late getting into the cloud game, and is threatened by the rise of Rackspace (RAX), lead sponsor of the Open Stack cloud infrastructure project. But it's still a powerful tech player, with sales running over $60 billion/year. The key question will be whether Dell is just selling Red Hat's offerings, or whether it's going to be an active contributor.
At the end of the day, it's developer buy-in that's at issue. Cloud platforms are not complete. It's kind of like windowing operating systems in the mid-1980s, which except for the Apple Macintosh remained vaporware for years. The more people you can get working on your version of the problem, the faster you come to a solution.
Dell could prove a big "get," or it could prove a boat anchor. Much depends on the resources that company is willing to devote to Red Hat's software production, and how it fits in with both Red Hat's own people and its larger developer community. If it proves to be a win-win over the next six months, it could encourage other big enterprise players to join in.
Disclosure: I am long MSFT.