$\left\lfloor x\right\rfloor$ and $[x]$ are the same concept? The function $f(x)$ integer part of $x$ is defined as the largest integer less than or equal to $x$. Generally I have always use this symbol to indicate the integer part of $x$
$$f(x)=[x]$$
instead of 
$$f(x)=\left\lfloor x\right\rfloor.$$
Are there some differences or are they represent the same quantity?
 A: $[x]$ is the older notation for the largest integer that does not exceed $x$. Earliest Uses of Various Mathematical Symbols says

Greatest integer function (floor function). Until recently $[x]$ has been the standard symbol for the greatest integer function. According to Grinstein (1970), "The use of the bracket notation, which has led some authors to term this the bracket function, stems back to the work of Gauss (1808) in number theory. The function is also referred to by Legendre who used the now obsolete notation $E(x)$." The Gauss reference is to Theorematis arithmetici demonstratio nova. Werke Volume: Bd. 2 p. 5.
Recently the symbol $\lfloor x \rfloor$ has come into use. It was introduced in 1962 by Kenneth E. Iverson who also coined the name floor function. [...]

$\lfloor x \rfloor$ has two advantages: it is completely unambiguous (there is nothing else it is used for) and it has a mnemonic symmetric partner in $ \lceil x \rceil $, the ceiling function. Examples of use:
$$ \begin{align}
 \lfloor 2 \rfloor &= 2 & \lceil 2 \rceil &= 2 \\
 \lfloor 2.5 \rfloor &= 2 & \lceil 2.5 \rceil &= 3 \\
 \lfloor -1 \rfloor &= -1 & \lceil -1 \rceil &= -1 \\
 \lfloor -1.5 \rfloor &= -2 & \lceil -1.5 \rceil &= -1 \\
 \lfloor \pi \rfloor &= 3 & \lceil \pi \rceil &= 4
\end{align} $$
