Those terms are often used interchangeably. However, there is a clear distinction between
Cumulative distribution function, or cumulative probability distribution function
and the probability density function.
The cumulative distribution function is the P(X \leq x), the probability that a random variable X assumes a value less than or equal to x. For continuous random variables, the probability density function is the derivative of the cumulative distribution function.
Sometimes people use probability distribution function for discrete random variables where you can actually talk about the probability for an elementary outcome (unlike continuous random variables where you have to talk about probability on an interval). The term probability density function is NOT used for discrete random variables. For discrete random variables, people often use the term probability mass function.
Myself when I use the term distribution function, I mean the cumulative distribution function (for both continuous and discrete RVs). So you have to decide from context how it is being used.