Without getting into the debate as to whether choosing to improvise in an audition is worthwhile or not, I would say the following: to a certain extent, it depends what you are auditioning for, and what you mean by delivering a speech.
Essentially, using a monologue as a test of talent is most prevalent in stage circles; it is pretty uncommon as an audition technique for screen, where the primary interest is in seeing what you can do with the given script for the project, whether this is learnt beforehand, or cold read.
Asking actors to perform monologues seems to be less popular in stage auditions than it used to be; at very least, most auditions will ask for a monologue coupled with a cold reading of the part that is actually being auditioned for. But monologues do still have their uses from a theatrical point of view; they can demonstrate something about how effectively an actor can use space, how convincingly they imagine the 'circumstances' of a scene around them, what kind of emotional journey they can take an audience on etc. All of these attainments are useful for a stage actor, and so, 'testing' the actor for such qualities is important. In a sense, it *doesn't* matter what gets presented so long as the intentions and the drive within the scene are coherent, although, in many stage circumstances, I wouldn't advise deviating too far from any set text. Classical work, for example, is often directed by those who are highly conversant with the text, and who demand close attention to it - many Shakespearean directors, for instance, literally wince when they hear verse lines being misdelivered. There are some good reasons for this attitude - for the truth is that subtext in stage work is revealed only through the words, through what the actors say to one another: we are never close enough to the actors to simply see a subtext register in their eyes/faces/etc. The story is told through the words (as well as gesture), and so, proving you can pay attention to the words can be very useful in auditioning for stage productions.
When auditioning for screen, it is a totally different scenario. The interest here is primarily in terms of reaction, spontaneity and believability, because the camera can detect the most minimal of subtexts in the change of a facial expression or a glint in the eyes. I have it on good authority from top casting directors that no-one actually cares what words you deliver at a screen audition, so long as they are in the ballpark, and delivered with the right conviction. Ad-libs are more than welcome if they flesh out an underwritten character. Any phraseology that doesn't suit your delivery, but which you can convert into a form that works for you, is more than acceptable. What you want above all is to come across as immediate and 'real'. In screen scenarios, improvisation will almost certainly help your cause, although it's very unlikely anyone will be asking you to improvise around a pre-prepped monologue - they *will* often ask you to improvise around the script as written - it's always assumed that the script, as given, isn't set in stone, anyway, and is subject to alteration if a line doesn't work, sounds clunky etc.