What do we gain by having "dependent types"?
Expanding my comment: Dependent types can type more programs. "More" simply means that the set of programs typable with dependent types is a proper superset of the programs typable in the simply-typed $\lambda$-calculus (STLC). An example would be $List_{2*3+4}(\alpha)$, the lists  of length $10$, carrying elements of type $\alpha$. The expression $2*3+4$ is at the same time a program and part of a type. You cannot do this in the STLC. 

The key rule that distinguishes dependent from non-dependent types is application:

$$
\newcommand{\TYPES}[3]{#1 \vdash #2 : #3}
\newcommand{\SUBST}[2]{\{#1/#2\}}
\frac{
   \TYPES{\Gamma}{\color{red}{M}}{A \rightarrow B} 
      \qquad
   \TYPES{\Gamma}{\color{red}{N}}{A}  }{
 \TYPES{\Gamma}{\color{red}{MN}}{B}   }
      \qquad
\frac{
   \TYPES{\Gamma}{{\color{red}M}}{\Pi x^A. B} 
      \qquad
   \TYPES{\Gamma}{\color{red}{N}}{A}  }{
 \TYPES{\Gamma}{{\color{red}{MN}}}{B\SUBST{{\color{red}N}}{x}}   }
$$

On the left you have the STLC, where programs in the premises 'flow' only  into the program of the conclusion. In contrast, in the dependent application rule on the right, the program $N$ from the right premise 'flows' into the type in the conclusion$^1$. 

In order to be able to parameterise types by programs, the syntax of dependent types must be richer, and to ensure that types are well-formed we use a second 'typing system' called kinds that constrains the types. This kinding system is essentially the STLC, but "one level up".

There are many explanations of dependent types. Some examples.


Dependent Types at Work, by Bove and Dybjer.
Dependent Types, by Aspinall and Hofmann.
Dependently Typed Programming in Agda, by Norell and Chapman.
Lambda Calculi with Types, by Barendregt.




$^1$  In terms of colours: with non-dependent types, black expressions in the conclusion are constructed from black expressions in the premises while
red   expressions in the conclusion are constructed from red expressions in the premises. With dependent types the colours can be mixed by having black parts of the conclusion being constructed from red and black parts of the premise.