Why use the control sequences \bigl, \biggl, \bigr or \biggr, as I can always use \big or \bigg?

As has already been pointed out in the other answers, using l ("ell") immediately following \Bigg, \bigg, \Big, and \Big informs TeX that the subsequent "fence symbol" -- (, [, \{, etc -- is to be given math-class "Math Open" rather than "Math Ordinary".

The consequences of this difference in the math class status of the fence symbol ("Open" vs "Ordinary) are particularly striking if the first math atom that follows the fence symbol is an arithmetic operator such as + (plus), - (minus), or \times:

• If the fence was assigned class "Math Open", i.e., if \Biggl, \biggl, etc was used, TeX will -- correctly -- not insert a bit of extra space between the fence symbol and the -/+/\times symbol, resulting in the symbol (correctly!) be typeset as a unary operator.

• Absent the l ("ell") qualifier, the fence is assigned class "Math Ordinary", and TeX will incorrectly interpret the -/+/\times symbols as binary operators (class "Math Bin") and thus also insert a bit more whitespace between these symbols and the next math atom.

The following examples illustrate the resulting differences in spacing, both between the opening fence and the arithmetic operators and between the arithmetic operators and what follows.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
\Biggl[-2x - 4y\Biggr]& \quad\text{\emph{with} ``l'': } 
     \begin{array}{l}
       \text{\textbullet\ tight spacing after opening fence;}\\
       \text{\textbullet\ first ``$-$'' symbol treated as unary operator}
     \end{array}\\
\Bigg[ -2x - 4y\Bigg]&  \quad\text{without ``l'': }
     \begin{array}{l}
       \text{\textbullet\ loose spacing after opening fence;}\\
       \text{\textbullet\ first ``$-$'' symbol treated as binary operator}
     \end{array}\\
\biggl(+3u + 7v\biggr)&\\
\bigg( +3u + 7v\bigg) &\\
\Bigl\vert \div 2x \times 4y\Bigr\vert &\\
\Big\vert  \div 2x \times 4y\Big\vert  &\\
\bigl\{ \times 3u \div 7v\bigr\} &\\
\big\{  \times 3u \div 7v\big\}  &
\end{align*}
\end{document}

\big (and its friends) create an Ord(inary) atom. \bigl and \bigr (and their friends) create Open and Close atoms. Spacing between atoms can vary.

For example, TeX inserts a thin space between an Op (big operator) atom and an Ord atom, while it inserts no space between an Op atom and an Open atom.

Compare (Op and then Ord):

\[ \sum\big( \]

with (Op and then Open):

\[ \sum\bigl( \]

You can see the a list of the 13 kinds of atoms in page 158 of the TeXbook. The table in page 170 shows the spacing between pairs of adjacent atoms.

EDIT: In the example give by bellochio, there's no difference in spacing due to the kind of atoms involved in both expressions; the only part which could be considered problematic is the initial part:

\biggl({\partial^2\over \partial x^2}

versus

\bigg({\partial^2\over \partial x^2}

However, using \showlists for the first expressions produces (only the relevant parts are shown):

### display math mode entered at line 1
\mathopen
.\hbox(14.5001+9.50012)x7.36115
..\mathon
..\hbox(14.5001+9.50012)x7.36115
...\hbox(0.39998+23.60025)x7.36115, shifted -14.10013 []
...\vbox(14.5+0.0)x0.0
...\hbox(0.0+0.0)x0.0, shifted -2.5
..\mathoff
\mathord
.\fraction, thickness = default
.\\mathord
.\.\fam1 @
.\^\fam0 2
./\mathord
./.\fam1 @
./\mathord
./.\fam1 x
./^\fam0 2

so the first atoms are of type Open, and Ord. For the second expression \showlists produces:

### display math mode entered at line 1
\mathord
.\hbox(14.5001+9.50012)x7.36115
..\mathon
..\hbox(14.5001+9.50012)x7.36115
...\hbox(0.39998+23.60025)x7.36115, shifted -14.10013 []
...\vbox(14.5+0.0)x0.0
...\hbox(0.0+0.0)x0.0, shifted -2.5
..\mathoff
\mathord
.\fraction, thickness = default
.\\mathord
.\.\fam1 @
.\^\fam0 2
./\mathord
./.\fam1 @
./\mathord
./.\fam1 x
./^\fam0 2

so the first atoms are of type Ord, and Ord. According to the table in page 170, TeX inserts no space between an Open and an Ord atom and also between two Ord atoms and that's why there's no difference in the spacing.

In that case, yes, perhaps, but try

$\log\bigl($ vs. $\log\big($

they are not the same, because \big( is AFAIR not of type math open

there are also examples regarding the type of the right fence (though I've forgotten about it at the moment)