These two concepts are so strikingly different that it is really difficult for me to understand how they can be confused. The confusion may stem from the ambiguous colloquial description you quoted (link?) which can be easily eliminated by presenting them with rigorous mathematical notations.
Short Answer: A probability space is a triple $(\Omega, \mathscr{F}, P)$, while a probability distribution is a (set) function. So they are very different.
Longer Answer: A probability space is a triple $(\Omega, \mathscr{F}, P)$ that contains a sample space $\Omega$, a $\sigma$-field $\mathscr{F}$, and a probability measure $P$, while a probability distribution (of a random variable $X$ that is a measurable function from $\Omega$ to $\mathbb{R}$) is a probability measure (i.e., a set function) defined on another measurable space $(\mathbb{R}^1, \mathscr{R}^1)$ where the random variable $X$ takes value on.
More Elaborations: As I made it clear above, to make sense of the term "probability distribution" (let's denoted it by $\mu$), a random variable $X$ must be defined in advance on the probability space $(\Omega, \mathscr{F}, P)$. In particular, $\mu$ is different from the original probability measure $P$, but it is a probability measure induced by $P$ (sometimes called "push-forward measure") as follows: \begin{align*} \mu(A) = P(X^{-1}(A)) = P(\{\omega \in \Omega: X(\omega) \in A\}), \quad A \in \mathscr{R}^1. \end{align*} Here $\mathscr{R}^1$ stands for the Borel field generated by open sets in $\mathbb{R}^1$. For this reason, $\mu$ is usually denoted as $P\circ X^{-1}$ or $P_X$. By any means, $\mu$ is a set function, which is obviously a different object compared to the probability space.
For completeness, I am quoting the formal definitions of these two concepts from an authoritative mathematics textbook (Probability and Measure (3rd edition) by Patrick Billingsley):
(Probability space) If $\mathscr{F}$ is a $\sigma$-field in $\Omega$ and $P$ is a probability measure on $\mathscr{F}$, the triple $(\Omega, \mathscr{F}, P)$ is called a probability measure space, or simply a probability space.
(Probability distribution) The distribution or law of the random variable is the probability measure $\mu$ on $(\mathbb{R}^1, \mathscr{R}^1)$ defined by \begin{align*} \mu(A) = P[X \in A], \quad A \in \mathscr{R}^1. \end{align*}
Comments on your Quotation:
A probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.
In my opinion, this sounds more like a description of $P$ in the original probability space, instead of $\mu$ on $(\mathbb{R}^1, \mathscr{R}^1)$.
It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).
Again, this "It" is better (if I have to) interpreted as $P$ -- accordingly the sample space and subsets of the sample space in this quotation correspond to $\Omega$ and $\mathscr{F}$ respectively.
In summary, I think the author of your quotation had confused "$P$" and "$\mu$" -- when he/she drafted this paragraph, he/she probably had $P$ in his/her mind and tried to elaborate it in words, for the random variable $X$, which is a prerequisite to define the term distribution, looks completely slipped away.
