For every question, there is separate argument why the general notion of compactness is better. I guess that the final definition is a result of gradual improvements. It is hard to explain it simply. So let's go through some examples.
A simple statement of the property is "closed and bounded".
Compactness should be a property of space, not a property of a set within a space. This is the reason why the definition "closed and bounded" fails. For instance the set of rational numbers in interval $[0, 1]$, or the interval $(0,1)$ are also topological (also metric) spaces. They are bounded and although they are not closed in $\mathbb R$, they are closed in themselves.
Instead of closed sets, we have to use another notion of metric spaces, completeness.
And we are not done, yet. There is still a problem with the boundedness. You can define another metric on the real line, $\rho(x, y) = \min(1, |x-y|)$. This is still a correctly defined metric on $\mathbb R$ and it produces the same topology, so if $\mathbb R$ is not compact, the modified space should still not be compact.
But the space happens to be bounded. So instead of bounded space, we need another notion, totally bounded space.
Now, it finally suffices. Actually, a metric space is compact if and only if it is complete and totally bounded. But
- the standard definition is much simpler,
- the notion of completeness and total bounded space requires the metric, while the standard definition uses just topology -- open sets. So the standard definition is more general.
i.e. Every sequence has a convergent subsequence.
Sequences are bad in general topology. The reason is that you are used to a fact that whenever there is a point $x$ in a closure of a set $S$, then there is a sequence from $S$ converging to $x$ in metric spaces. This is not true in general. For example, you can take the space $\mathbb R$ and add a point $\infty$ (I do not mean usual $+\infty$) such that $\infty$ is in a closure of a set $S$ if and only if $S$ is uncountable. Then there is no sequence of real numbers converging to $\infty$ although $\infty$ is in the closure of real numbers.
There are more examples why sequences are bad but they are often more complicated and require set theory. An example of space which is not compact but every sequence has a convergent subsequence is omega 1
On the other hand, there is an equivalent definition of compactness similar to the Weierstrass Property: For every infinite set $S$ of size $\kappa$, there is a $\kappa$-accumulation point $x$. It means that every open set containing $x$ contains $\kappa$ elements of $S$.