How does the 1999 film Hamlet by the Royal Shakespeare Company differ from the play Hamlet written by William Shakespeare?There are many films made...

Your question isn't as simple as it might first appear to
be.  The main reason for this is that, technically, this is a "filmed" version of
Hamlet, but it is performed by a theatre company -- The Royal
Shakespeare Company -- and produced for television entirely on a soundstage that
simulates the environment of an indoor stage production.  This version lacks the
switching between Interior and Exterior locations more common in an actual film
adaptation of the play.  (It was adapted for televison in 2009 for airing in 2010, not
in 1999.)

This sort of re-staging of a theatrical
production for the camera from different angles and with different camera placements
does make the play read like a "film," but this production falls more closely into line
with the made for television series for the BBC called "The Shakespeare Plays" that were
made in the 70's and 80's.

It is important to establish
that this is a filmed version of a stage play, because it affects your question, making
it less possible to contrast it with the play itself since it is the play.  The choices
of costuming and setting are not "changes" to the script of the play, since every play
must have these elements chosen for production.

One of the
elements that I see that would allow you to contrast this filmed version of the play
with the one presented theatrically, is that there are camera angles that allow the
audience to perceive only what is chosen in any given moment.  For the audience in a
theatre, where to look is a choice that each viewer makes, not something chosen for him
or her by a camera.  So, for example, in the opening scene, Barnardo is shot walking his
post as guard from above, so that it appears to be some sort of surveillance footage. 
This effect would definitely not have been possible
onstage.

Another obvious difference is that the soliloquies
that Hamlet shares with the audience in a live production when he is alone onstage, here
lack that direct connection to the audience.

For the most
part, this is a recreation of a staged production of Hamlet, and
so, in the choices of action, costuming and setting, is not possible to contrast with
the play since it is, in fact, the play itself.  I have provided a couple of links to
more information on the transferring of this stage production to the TV screen
below.