We use closest representations for both of them, but they are not completely true.
$\frac{22}7$ and $3.14$ are not exactly $\pi$ but we use them as the best option available.
$\frac13$ is $0.\bar3$ but that can be $0.333$ or $0.333333$ and these are not equal.
So why is one irrational and other is not?
On
The word irrational means the number cannot be expressed in a ratio of two integers. $\frac{1}{3}$ can obviously be expressed as a ratio of two integers. For $\pi$ and other irrational numbers however, there are no integers $a$ and $b$ where $\frac{a}{b}$ will ever equal that number.
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Having an infinite decimal expansion is not what makes a number irrational. A rational number is any number that can be expressed as a fraction - that is, the words rational number and fraction are essentially synonymous.
$\pi$ is irrational because it cannot be expressed as a fraction - $22/7$ is a close approximation, but no fraction can ever exactly represent $\pi$. On the other hand, $1/3$, being a fraction, is rational by definition, irrespective of any infinite repeating decimal sequence.
The connection between irrational numbers and decimal sequences is this - if a number is irrational, it's decimal sequence cannot terminate, and furthermore the decimal sequence cannot be periodic, or repeating. This doesn't mean there can't be any pattern in the digits, just that they can't repeat themselves endlessly in uninterrupted fashion. A rational number, on the other hand, can have either a finite decimal expansion or an infinite expansion, but if the decimal expansion for a rational number is infinite, then it must be periodic, or repeating.
So the properties of the decimal expansions of rational and irrational number are a consequence of their definition in terms of representability by fractions, and not a definition in and of themselves.
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"Rationality" is not about closeness of approximations - we use approximations for everything! If a person weighs $151.6629943$ pounds, they'll probably say they weigh $151$. And $151.6629943$ doesn't even go on forever!
A number is "rational" if it is a whole number divided by another whole number. Repeating decimals can always be written that way - if the repeating section is $a_1a_2\cdots a_k$, then the number can be written as $\frac{a_1a_1\cdots a_k}{99\cdots 9}$. Importantly, this fraction representation is required to be exact, unlike the decimal approximations $0.333$ and $0.3333333$ you were talking about.
$\pi$, on the other hand, cannot be written this way. $\frac{22}{7}$ is a good approximation, but it is not exact; in fact, there is no fraction that would exactly equal $\pi$.
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Repeating decimals are simply geometric series which add up to
$$a + ar + ar^2 + \cdots = \frac {a}{1-r} $$
where $a$ and $r$ are rational numbers, so the result is rational.
For example:
$$\begin{align} 0.23\,23\,23\ldots &= 0.23 +0.00\,23+0.00\,00\,23 +\cdots \\[2ex] &= 0.23 + 0.23\left(\frac1{100}\right) + 0.23\left(\frac1{100}\right)^2 + \cdots \\[2ex] &= \frac{0.23}{1-\frac1{100}}=\frac {23}{99} \end{align}$$
Something is irrational if its decimals go on forever and do so with no pattern. The number $\pi$ fits both of these criteria; however $1/3=0.333\cdots$ does not fit the second criterion because its digits repeat.
If numbers exhibit a pattern, it’s a clue to us that our decimal (base $10$) system represents them cleanly—and sure enough, when we look at the math, we see this is the case!