Conditional Probability and Causality
The idea that you can define causation in terms of conditional probability was the 'probabilistic causality' programme in philosophy associated with e.g. Cartwright and Eels. Arguably, it failed (See Pearl for the argument). A good introductory read on the topic is here. Several counterexamples to the probability raising relationship you suggest are in section 2.10.
As a result, it is unlikely that you will fully understand or otherwise reconstruct the difference between correlation and causation using only the machinery of conditional probability because it is insufficient. Explicitly causal i.e. non-probability assumptions are needed in addition.
Correlation is, as @Michael Chernick and other commenters point out, closely related to conditional probability. In a narrow technical sense it is a standardised undirected measure of linear or at least monotonic association between two variables. In a wider informal sense it is as Michael describes it: a departure from statistical independence. In either sense it may appear as a result of an underlying causal relationship. Or not, e.g. it instantiates one of the counterexamples above or exhibits Simpson's Paradox. Hence the difficulty/impossibility of reconstructing the one with the other.