I find it easiest to answer this question with an example.
All these systems deal with our allocation of resources. So we can do a basic comparison by simplifying society into a single resource, and analyse its acquisition, allocation, and application in regards to supplying a single need.
Yes it's simple and society is a lot more complicated, but many people claim they can explain capitalism and communism in a single sentence, so that's what we're going to do, just for the purpose of explanation.
Lets take wood as the resource and consider its use in keeping us warm. Imagine a deep forest, and a small society within it, living in huts that they need to heat.
I'll start with the "elevator speech" version, which you can take with you.
- In capitalism, we all fight over the wood, some people end up getting all the wood and then sell it back to us.
- In communism, everyone gets an equal amount of wood.
- In a Resource Based Economy, we build a solar thermal farm and heat everyone's hut .
Let's explore it in a bit more detail.
It begins, (in Ayn Rand's dream world), with everyone having an equal chance of acquiring enough wood to feed their family. (Or warm them!) There is plenty of wood to go around, you just have to work to get it. The more you work, the more wood you can obtain.
If you work really hard, and create a surplus, you can then use this extra wood to trade for tools to cut down even more wood, giving you an advantage over others. Accumulation perpetuates itself - the rich get richer.
We could of course complicate the analogy by going into how automation or third world labour is taking all the jobs and leaving the first world poor and unemployed, and all the issues that society is currently facing, but let's keep it simple - the CAPITALISTS (those with the capital) gained overall control over the resources and their allocation. (We don't even need to bring government into this!)
Obviously there are many differences between communism and socialism, and I guess you could come up with separate analogies for them both if you wanted, but for simplicity, I'm going to bundle them into one.
In your communist forest, everyone works, and everyone is given enough wood to keep their family warm. Actually, that's the socialist forest. In a communist forest, the government elites take more than everyone else and the people are left to share out what's left (which will be less than the required amount).
This analogy is a lot simpler than capitalism, mainly because there is a lot to be said about capitalism.
Resource Based Economy
Now, how does it work in a Resource Based Economy? How do we share the wood? Who decides who shares the wood?
Well, in a Resource Based Economy, we all come together for a short time, build everyone extremely high tech, long lasting, efficient, houses and apartments with solar panels and recreation centres. Cities with automated transportation and agriculture. Once built, and everyone is looked after, we can all do the jobs we choose, improving and maintaining the system we've built.
Of course it's not that simple, but the idea is. Instead of making the resources a free for all, or dictating from the top how they get allocated, we use them in the most efficient way possible, in accordance with the latest science and technology, in order to provide the most benefits for society as a whole.
[Edit:] The main point is that this is a totally different approach to capitalism and communism. It's a collective system, for sure, but it's a system based around technology, efficiency, environmental and social good, whereas capitalism is centered around irresponsible growth. The unit by which success of the system is measured is radically different than capitalism or communism.
In the end, the other methods only ever benefit an elite few. And realistically, those benefits are a false economy. When we're all working together for the benefit of everyone, we can create more benefits than even the elite of today could dream of.