Stationarity vs weak dependence
Weak stationarity and weak dependence are complementary conditions.
A weakly stationary time series $y_t$ has an underlying statistical process which is time-invariant.
This is characterised by three conditions:

$$E(y_t)= \mu$$
$$E((y_t-\mu)^2)=\sigma^2 &lt;\infty$$
$$Cov(y_t,y_{t+s})=Cov(y_t,y_{t-s})$$

Condition (1) indicates that a weakly stationary series has an underlying (or long-run equilibrium) unconditional average.
Condition (2) indicates that the unconditional variance of a weakly stationary series does not change through time and remains within a certain range/finite.
Condition (3) indicates that a weakly stationary series has stable (symmetrical) dynamics between periods, which only depend on the number of periods, rather than time itself.
That is to say, Condition (3) states that a weakly stationary process we can assume a stable relationship between certain periods, regardless of whether it's the year 2000 or the year 3000. Time plays no role, only the relationship across periods.
Weak Dependency:
Weak dependency ensures mean reversion back to an underlying process.
Weak dependency states that the correlation between $y_t$ and future periods $y_{t+h}$, tends to zero sufficiently quickly. And with periods increasing to infinity, dependency should tend to zero.
$$Corr(y_t,y_{t+h})\rightarrow 0, h\rightarrow\infty$$
Importance in the Context of Time Series
With time series data, our central objective remains to estimate the magnitude of change in Y terms of linear unidirectional causation from X.
That is to say, our goal is to take a sample time series and fit a line which best explains the change in Y given X characteristic of the underlying population.
Yet, in the time series context, conducting estimation representative of the underlying population is undermined by a failure of i.i.d conditions
An evolving (non-stationary) time series, entails that observations will not be drawn from an identical distribution and parameter estimates will not convey a constant linear (time-invariant) relationship on Y given X;
Furthermore, given that time series data is necessarily non-independently distributed (i.e. current realisations dependent on past realisations), then again parameter estimates will no longer convey information on the underlying population.
It follows that two mutually dependent properties must be upheld to ensure the i.i.d conditions known as:

Sufficiently fast weak dependency – random variables should exhibit
finite memory/asymptotic independence;

Weak stationarity – the underlying statistical properties (mean,
variance, and covariance) should be time invariant as if drawn from
an identical distribution.


Note that both conditions are mutually dependent, given that sufficiently quick finite memory is a requirement for a process to exhibit mean-reversion back to some underlying population.
Moreover, the combination of weak dependency and weak stationarity ensure that a property known as ergodic stationarity is upheld.
Essentially, ergodic stationarity implies that with a quick return to an identical distribution, then realisations are representatives of the greater process, and averaging is relevant.
Further, weak stationarity is important in terms of the exogeneity condition to control for a causal relationship, given that if contemporaneous exogeneity holds in one period; then by weak stationarity, it should hold for all periods.
Also, ergodic stationarity ensure that the process mimics a random independent sample required to uphold LoLN/consistent estimation/CLT are valid.
Finally, weak stationarity overall avoids the problem of so-called spurious regression in which two mutually trending series are wrongly identified to have a causal relationship.