f(t) gives the probability as a point on a distribution curve while F(t) is the area beneath the distribution curve to the left of the point defined by f(t), representing the sum of the probabilities up to the point f(t).
A typical example is a normal distribution curve (bell curve). Along the t axis we may have the discrete (non-continuous) values, 0, 1, 2, 3, etc. The curve shows the probability of 0, 1, 2, 3, ... successes or hits. (In fact, the curve is really a series of blocks, one block for each value of t). So we have f(0), f(1), etc., for individual probabilities, t=T. F(t) gives the sum of the probabilities up to and including a particular value t=T, i.e., t<T.
When t is continuous (it can take random values in between discrete integer values so t>0), as indicated in your question, the same rule applies to PDF (individual probability) and CDF (cumulative probability). The curve demonstrates or illustrates that t is continuous.