Let me quote some biographical details:
--------------------------------------
Galileo Galilei's parents were Vincenzo Galilei and Guilia Ammannati.
Vincenzo, who was born in Florence in 1520, was a teacher of music and
a fine lute player. After studying music in Venice he carried out
experiments on strings to support his musical theories. Guilia, who
was born in Pescia, married Vincenzo in 1563 and they made their home
in the countryside near Pisa. Galileo was their first child and spent
his early years with his family in Pisa.
-------------------------------
Based on this information his lastname is Galilei and his firstname is Galileo.

Why do we refer to him as Galileo instead of Galilei? I never saw in
official science textbook to refer to Albert Einstein as Albert?!

In Europe except England they refer to him as Galilei.

What is the reason of this discrepancy and which one is correct?

It's quite common for historical figures to be mentioned by their
first names. Rembrandt, Napoleon, Dante, and Leonardo come to mind.

....... and Jesus.

....... and Bryan. :)

I've often wondered the same thing about Saddam Hussein. Why do so
many people [English-speaking people at least] call him "Saddam"
instead of "Hussein"? I know Hussein is a common surname, but we would
call a John Smith "Smith" despite his surname being common. Hmmm.

Saddam Hussein isn't <firstname> <lastname>.  Hussein is his given
name and Saddam is a title he adopted for himself - it means something
like "The Crusher".

Ian G.

Re Saddam Hussein: Interesting! Thanks, Ian G.
---IndexTurret

Adding to pinkfreud's list: Michelangelo (last name Buonarroti).

Trust Pinkfreud-ga.
Till I got to Europe, I didn't know he had any other name.

There is a cutoff date (and I forget exactly when, but basically at
the start of the baroque period) where one starts calling European
historical figures by their last names. Michaelangelo Buonorotti is
referred to by his first name, but Michaelangelo Caravaggio is
referred to by his last.

It gets complicated for cultures that base a 'last name' on a
patronymic, as with Arabic (or, in the case of Icelandic, on
matronymics as well), and one ends up digging thru a style guide (such
as that of the Modern Language Association). Languages such as Spanish
or Portuguese that create a doubled last name based on both last names
of the parents are also complicated.

Waukon,
You may be right about that cut-off date, although Caravaggio may have
just "suffered" lack of first name importance because Michaelangelo
hat blocked that name.
In Iceland, girls have the name of their fathers with -dattir 
(daughter) as an addition, so it is not a matronym.  I believe that in
Sweden into the 19th century, this was also still known in some areas.

In Iceland, girls have the name of their fathers with -dattir 
(daughter) as an addition, so it is not a matronym. 

I knew better and stand corrected. I actually don't know of any
culture that forms 'last names' as a matronymic.

I remember my Norwegian grandmother telling me how her mother was once
known as Ragnhild Mattiasdottir (or something like that, forgive me
any misspelling). Not just Sweden in the 19th century. but also Norway
in the vicinity of Lillehammar.

Your grandmother too!
But we may be getting to matronyms with all the single mothers around,
and in Germany girls can keep their maiden name, and then the parents
can pick which name the children will have (but after the first one,
they all have to have that name).