Cream Cheese vs Yogurt Cheese: what / how much difference does the bacteria make?
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.