If you have raw milk and let it sit, the cream will indeed rise to the top. To separate, you can just wait and skim off the cream as you did. However, if you store the raw milk in the refrigerator, it will take longer for the cream to rise. Perhaps that is why you are having difficulties. Alternatively, you can use a spigot jar to drain the "skimmed" milk from the raw milk, leaving the cream behind.
I think you end up with about 1/10th of your raw milk that is cream.
Just as important as the bacterial culture is the use of rennet in cream cheese, which aids in the removal of liquid whey. When making cream cheese, the point is to drain much of the whey, resulting in a semi-solid texture. Rennet helps encourage the solids to curdle and squeeze out liquid. Yogurt doesn't necessarily include the draining step, though it can be done if you're looking for a thicker Greek-style yogurt. In this case, the acid produced during fermentation while making yogurt aids curdling and helps produce the final texture.
In fact, it's possible to take the draining process even further with yogurt, resulting in what's often called yogurt cheese or labneh. The final texture can vary a bit depending on how long it's drained, and whether you use weights to encourage the process.
In my experience labneh still never gets quite as solid as cream cheese, but it's pretty close when sufficiently weighted and drained. Labneh also retains yogurt's tangy flavor, which is mostly an effect of the bacterial culture. Though I haven't measured, I would expect that the pH of labneh is lower, so it's probably not always appropriate as a direct substitution for sensitive applications like baking. In other places, you could definitely use labneh instead of cream cheese. If you're having difficulty locating cream cheese cultures, this would be the easiest tack to take.
I haven't tried (or seen) both rennet and yogurt culture used together, but my suspicion is that it would take the curdling action a bit too far for the result to be smooth and spreadable.
So, tl;dr: the major difference is that yogurt culture is calibrated to produce a higher level of acid, resulting in a tangier flavor and reducing the need for rennet to curdle the solids. If you follow the same procedure, varying only the culture used, you'll have a reasonably similar end result.
Best Answer
I am skeptical that butter from yogurt is a thing.
When yogurt is made the milk proteins denature and form a mesh that traps all the large molecules in the milk. Water, sugar, and some small molecules can come out but the fat never does- it's huge and tightly bound up in the gel.
Even when yogurt is blended up the whey will separate out but the fat never does. I have a hard time believe that extricating the fat from that protein mesh will be easier than just letting it rise to the surface in milk.
Perhaps thoroughly cooking the yogurt would melt out the fat but you would still lose the rest of the milk solids and would have ghee.
With purchasable yogurt it would be even less feasible because it is often made from low-fat milk to begin with.
EDIT-
Searching online I was able to find Indian recipes for making butter from yogurt: http://www.youtube.com/watch?v=_vK8hW_oSu0
In this case they made yogurt from heavy cream. The goal was a cultured butter- not easier butter.