My reading is that they are not the same thing. The following paragraph from the Wikipedia page for "probability density function" may help or may just confuse you further:
The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. Further confusion of terminology exists because density function has also been used for what is here called the "probability mass function."
Your definition of a "probability function" ("assigns a probability to an event") seems more applicable to a probability mass function, $f(x) = Pr(X=x)$, which is a function that gives the probability that a discrete random variable is exactly equal to some value. A probability density function, maps a continuous variable to the relative likelihood of that value as compared to other values. For given $x$, the value of the probability density function $f(x)$ may have a value greater than 1, which is a no-no when it comes to probability. For example, the uniform distribution on the interval [0, .5] has a value of 2 on that interval and 0 elsewhere. In order to obtain probability from the probability density function, one must integrate between the values of interest. This means, interestingly, that the probability of a given specific $x$ occurring is technically zero. This is because if in integrate $f(x)$ from $a$ to $b$ where $a=b$ you get $F(a)-F(a) = 0$ where $F(x)$ is the cumulative distribution function.