Correlation doesn't PROVE causation.... ...but it bloody well DOES suggest it, at least in the course of our daily lives.
The reason this phrase is so catchy is that it's counter-intuitive, and easily proven to be true. People love to use it as a "gotcha" phrase, PRECISELY because in regular life correlation does in fact usually imply causation.
No, you have it exactly backwards. Causation usually implies correlation(*). But there are lots of correlations that are not in any way causally related, such as a) the decline in piracy and increase in global temperatures in the past two centuries; b) the rabbit population in Australia and the performance of the London stock exchange for the past century; or c) the monthly per capita consumption rate of ice cream and the monthly per capita rate of drowning deaths for seaside locations north of 40 degrees latitude. The first two are because both sets of observations have changed in consistent fashions over time. The third is because people seek/avoid both swimming and ice cream based on seasonal variations of temperature, i.e., there's what's called a "lurking covariate". In none of these cases does one of the sets of observations cause the other. You won't change global warming by encouraging piracy, swing the stock market by raising rabbits in Oz, or change drowning statistics by banning ice cream.
(*) Causation is not always associated with correlation, because correlation measures tendency towards a linear fit. If there's a non-linear relationship, you can have perfect causality and zero correlation. Example - Let X be uniformly distributed between -1 and 1, and let Y = X^2. If I tell you a particular X value you can predict Y perfectly, but if you work the math of correlation you'll find Corr(X,Y) = 0.