Probabilistic means there is uncertainty in the process where the possible outcomes of some event may or may not have fair (equal) shares. For example: When you throw two fair dice, the probability of getting a sum of the observed top faces greater then 4, is greater than getting a sum of less or equal to 4.
You can think of Random, as a special case of probabilistic where all the possible outcomes of some event in an experiment have equal probabilities of happening. Example: Drawing one card out of a standard 52-card deck gives $1/52$ probability for each of the cards (given no preference or prior information whatsoever about any of the cards). For more information about random chances, check Uniform Distribution. From the definition of uniform distribution: If an experiment is random, the probability of an event is the number of possible outcomes divided by the total number of possible outcomes.
To get back to the text you quoted:
random experiment is a probabilistic experiment.
That still holds, given that
random is a special case of
probabilistic, so every random experiment is probabilistic but the opposite is not necessarily true.
Note: The common distinction is not between
probabilistic, but between