In one word, Dirac delta function is for continuous cases, but Kronecker delta is for discrete cases.

Dirac and Kronecker delta functions are useful in random presses. We can always find them in papers and books especially when white noise appears. They usually are both denoted as $\delta(*)$. They are the same? No!!!

1. Dirac delta function:
$\delta(t)=+\infty$ if $t=0$

$\delta(t)=0$        if $t\ne 0$

Moreover, don’t forget the constraint: $\int_{-\infty}^{+\infty} \delta (\tau) d\tau=1$.
2. Kronecker delta function:
$\delta(k)=1$ if $k=0$

$\delta(k)=0$ if $k\ne 0, (k=\pm 1, \pm 2,...)$

The most famous property

$\int_{-\infty}^{+\infty} \delta (\tau-x) f(\tau)d\tau=f(x)$

is for Dirac delta function! That’s because integrals require continuous function, while Kronecker delta function is a discrete one. In fact, similarly we have

$\sum_{i=-\infty}^{+\infty} \delta (i-k) f(i)=f(k)$

Now you may understand why Kronecker delta function will have $\delta(k)=1$ but not infinity when $k=0$.

Moreover, for white noise $w(t)$,

$E[w(t)w(t-\tau)]=Q\delta(\tau)$

here is Dirac delta function.

For white sequence $w(k)$,

$E[w(k)w(k-\tau)]=Q\delta(\tau)$

here is Kronecker delta function. Note for discrete cases, we can also find $\latex w(t), w(t+1)$, remember it is discrete.