Not every wave function is a wave packet. Wave functions are frequently considered to include functions that are not properly normalizable, like $\operatorname{e}^{ikx}$, so we describe them as "normalized to the delta function." That is, we demand $$\delta(k - k') = \int_{-\infty}^\infty \operatorname{d} x \psi^\star(x, k') \psi(x, k),$$ leading to things like $\psi(x,k) = \operatorname{e}^{ikx} / \sqrt{2\pi}$.
With a wave packet we make the more strict requirement that it be normalizable and localized in both $x$ and $p$. The canonical example of a wave packet is:$$\psi(x) \propto \exp \left(-\frac{1}{4} \frac{(x - x_0)^2}{\sigma^2} + i \frac{p_0 x}{\hbar}\right),$$ which has $\langle x\rangle = x_0$ and $\langle p \rangle = p_0$, and is normalizable with finite width in both $x$ and $p$ space.
Another example of a wave packet is the $\operatorname{sinc}$ function:
$$\psi(x) \propto \frac{\sin\left(\frac{x-x_0}{2\hbar}\Delta p\right)}{(x-x_0)} \operatorname{e}^{ip_0 x / \hbar}.$$ It is localized in $x$, in the sense that it is normalizable and has definite quantiles, but it has divergent variance. In $p$ it is much better behaved because it is a boxcar function that stretches from $p_0 - \Delta p / 2$ to $p_0 + \Delta p / 2$.
