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The distinction is simple: The probability that an event will occur is the fraction of times you expect to see that event in many trials. Probabilities always range between 0 and 1. The odds are defined as the probability that the event will occur divided by the probability that the event will not occur.
A probability must lie between 0 and 1 (you cannot have more than a 100% chance of something). Odds are not so constrained. Odds can take any positive value (e.g. a !T probability is the same as odds of 2/1). If instead we use odds (actually the log of odds, or logit), a linear model can be fit.