The most important difference between the two worlds is the timing involved in the magic used. In Harry Potter spells are instant and the results are seen as soon as the caster works the spell. This is very different to real life magic which can take much longer to manifest itself. Its also interesting to note that all spells in Harry Potter (with the exception of potions) take the same amount of time and effort to cast. A complex spell in real magic may require several workings over a period of time to bring the results to fruition. Its inconceivable that both a spell to cause death and a spell to mend spectacles (neither of which would work in real magic) would require the same amount of effort.
Which brings me to my next difference - magic in the world of Harry Potter appears to be all about circumventing natural law. Physical repair, transforming human shape and calling objects to fly to your side are all impossible - genuinely and simply impossible. A real witch works within the constraints of natural law. I can no more summon my broom to me than you can. There are some caveats and exceptions to the natural law principle but in general witches cannot cast magic that breaks natural law.
Harry Potter magic is all about spells and results but the life of a real magical practitioner is very different. Keeping your spiritual health robust is a must for a witch and there are many practises - grounding, centering, shielding, aura cleansing - that are likely to make up the bulk of a witch's work. We are never granted an understanding of the work behind the magic in Harry Potter or any insight into how HP witches maintain their spiritual energies.
The classes of Hogwarts separate magic into very separate categories. Again real magic doesn't work that way. A single spell may involve specific herbs (HP Herbology) visualisation (HP Transfiguration), shielding to prevent negativity sent back from anyone involved in the working (HP Defence against the Dark Arts) and a quick check of the Tarot to check that the wind is in your favour (Divination). Real witches don't operate within different branches of magic, they incorporate all elements of what they need into the magic they undertake.
However, having said all that, there are some areas of overlap as well. JK Rowling explores the link between intent and results beautifully in the statement by Bellatrix Lestrange to the effect that in order for a curse to work, you have to really mean it. This is true of all spell work, intent isn't the be all and end all in magic but it is a key element. Especially in curse work, Bellatrix is right, you do have to summon sufficient anger and hate to come anywhere close to being able to cast an effective curse.
The Harry Potter world places a lot of emphasis on the wording of a spell and again this bears a close relation to real magic. Words express intent so it is important to word a spell carefully. Witches will often keep notes of spells that worked well. However the personal book (Grimoire) of the witch is different from the spell textbooks used by Harry and his friends in class. Many practising witches (myself included) won't use spells written by anybody else (and certainly never without knowing the writer or the source). A witch may use a spell book for ideas and inspiration but she is unlikely to copy an individual spell in its entirety.
Witches in both fiction and reality use wands but in Harry Potter the wand is suggested to have a power of its own. I have a wand (from which I get far fewer sparkles) but in the hands of another person my wand is nothing but a twig, it is a tool to direct the power that comes from me and not a power in itself.
Witches do use herbs and some of the herbs used in Harry Potter do exist and are used by real witches. The one that stands out to me is Mandrake. JK Rowling makes the effects of digging up Mandrake roots a bit more spectacular than they are in real life but the root of the Mandrake do resemble a person and historical record do show that people used to get their dogs to dig them up because of the myth that the scream could be fatal to a human (there isn't a scream of course). One of the hardest herbs to grow incidentally, I've never even bothered to attempt it, I buy mine dried.
Witches do make themselves invisible, not in the literal sense but in the blending into the background sense (which is why I rarely get asked to present my train pass...) Nothing as glam as an invisibility cloak I'm afraid, more a shielding technique. Closest in the books is the shield Hermione creates over the tent in book 7. The difference being that when a witch uses invisibility spells she is perfectly visible to the naked eye but her presence becomes less obvious and people go past without notice or comment.
So - many differences and some similarities. Magic in fantasy is more immediate and more dramatic than anything us real witches do. But whatever any of them say, we'd all secretly like to be able to shoot sparkles out of our wands!