Difference Between Dirac delta and unit impulse function
The Dirac delta $\delta(x)$ is a continuous argument (generalized) function where $x$ is a continuous variable and can be time, space, frequency etc. Unfortunately, a simple definition like

$$\delta(x) = 
\begin{cases}
\infty ~~~&amp;,~~~ x=0 \\
0 ~~~&amp;,~~~ x \neq 0
\end{cases}
$$

does or can not properly describe its behaviour. You need to define its properties under the integral sign, and that requires advanced mathematical stuff for a formal verification. But  a simplified Riemann definition of $\delta(x)$ is given as:

$$\int_{-\infty}^{\infty}f(x)\delta(x-a) dx = f(a)$$ for a sufficiently smooth test function $f(x)$. All properties of $\delta(x)$ follows from this Riemann (seeming) integral operation.

On the other hand, the discrete counterpart, the unit-impulse or the Kronecker delta is a sequence of integer index $m$ with very well defined behaviour as:

$$\delta[m] = \begin{cases}
1 ~~~&amp;,~~~ m=0 \\
0 ~~~&amp;,~~~ m \neq 0
\end{cases}
$$
These two functions serve the same purpose for discrete and continuous systems, hence share the same name impulse. Yet they are two different, distinct entities; the Dirac impulse is a very problematic concept, whereas the discrete  impulse is a mere simplicity.