Can anyone provide examples as to why choose one over the other? Are they calculated diferently?
Yes, they are very different.
Conceptually, a reflective measurement model happens when the indicators of a construct are considered to be caused by that construct. For example, an intelligence test: if you are more intelligent, you have a higher probability of getting the correct answer to a question. Hence your intelligence level is (theorized to) predict the score on the question. The latent variable (intelligence) is the predictor, the measured variable (test) is the outcome.
A formative measurement model happens when the measured variables are considered to be the cause of the latent variable. The value of a car is determined by its age, condition, size, make, etc. If a car is more valuable it does not turn from a Mercedes into a Ford. Instead, being a Mercedes is a predictor of being more valuable, and value is the outcome.
In a reflective measurement model, we expect the covariances between the indicators to be zero, when the latent variable is partialled out - that is the reason that two test scores correlate is because they are caused by the same thing.
In a formative measurement model, we don't have anything to say about the covariances of the items, they could be zero, positive or negative. Formative measurement models are harder to estimate - they are not identified on their own.