What is the difference between Witchcraft and Wizardry in HP?

Question

Hogwarts School of Witchcraft and Wizardry is the whole name of this majestic and well-known institution. Such institutions don't have names that don't make sense, or names longer than one would expect just for fun.
Because of that, I think, that there should be a reason why the name is this and not just i.e. Hogwarts School of Witchcraft. That means that Witchcraft and Wizardry aren't the same things.

What is the reason for the name? What is the difference between the Witchcraft and Wizardry?

(Possibly the out-of-universe reason may be that it just sounds better, but if so it's sad, so I hope, you'll come up with something else)

Answer 1

Witchcraft and Wizardry mean the same thing, the only difference being who is doing it. Simply: it's witchcraft when a witch does it and wizardry when a wizard does it. There's nothing more to it than that.

Wizard and its derivatives often functions as the generic he in the Harry Potter series. See for instance the name "the Wizarding World". There are also gender neutral expressions like "magical".

Hogwarts could have been titled with the generic he (Hogwarts School of Wizardry) or in a gender neutral way (Hogwarts School of Magic). But could doesn't equal must or even should.

Just as in the real world we don't make exclusive use of the generic he/gender neutral titles, neither does the Wizarding World. Much as you might say, 'greetings ladies and gentlemen' rather than 'greetings guests', the Wizarding World might say 'greetings witches and wizards'.

The titling of Hogwarts is no different and perfectly correct. We might speculate why they choose to name it so (my guess is to emphasise that this was a co-ed school which otherwise might get lost) but that would be speculation beyond your question.

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Some evidence for the use of wizard as a "generic he"

Wizard is used to refer to genderless magical things quite often. For example:

Money

Harry was turning over the wizard coins and looking at them. He had just thought of something that made him feel as though the happy balloon inside him had got a puncture. - HPPS

and

“They didn’ keep their gold in the house, boy! Nah, first stop fer us is Gringotts. Wizards’ bank. Have a sausage, they’re not bad cold — an’ I wouldn’ say no teh a bit o’ yer birthday cake, neither.” - HPPS

Laws

"Don’t you think it’s a bit odd,” said Harry, scrambling up the grassy slope, “that what Hagrid wants more than anything else is a dragon, and a stranger turns up who just happens to have an egg in his pocket? How many people wander around with dragon eggs if it’s against wizard law? Lucky they found Hagrid, don’t you think? Why didn’t I see it before?” - HPPS

Photographs (and notice this one explicitly includes a woman).

It seemed to be a handsome, leather-covered book. Harry opened it curiously. It was full of wizard photographs. Smiling and waving at him from every page were his mother and father. - HPPS

It is also used to refer to groups of both witches and wizards. For example here:

“Are all your family wizards?” asked Harry, who found Ron just as interesting as Ron found him.

“Er — Yes, I think so,” said Ron. “I think Mom’s got a second cousin who’s an accountant, but we never talk about him.” - HPPS

Derivatives are also common and used in a gender neutral way.

For example: Families

“I really don’t think they should let the other sort in, do you? They’re just not the same, they’ve never been brought up to know our ways. Some of them have never even heard of Hogwarts until they get the letter, imagine. I think they should keep it in the old wizarding families. What’s your surname, anyway?” - HPPS

Magical Society:

Quidditch, the most popular sport in the wizarding world (six tall goal posts, four flying balls, and fourteen players on broomsticks). - HPCS

Exams taken by both girls and boys

“Wish I knew what he was up to,” said Fred, frowning. “He’s not himself. His exam results came the day before you did; twelve O.W.L.s and he hardly gloated at all.”

“Ordinary Wizarding Levels,” George explained, seeing Harry’s puzzled look. “Bill got twelve, too. If we’re not careful, we’ll have another Head Boy in the family. I don’t think I could stand the shame.” - HPCS

Answer 2

These words have different connotations, and combined they represent the whole of the magic lore. The word 'witch' (and its male equivalent, warlock) both have clearly negative undertones and are associated with dealings with dark powers and curses, while the word 'wizard' (sage, wise man) has a much more positive meaning.

It is also intriquing that the word 'wizard' is a much later one (circa the XVth Century AC), which likely means that the original name of the Hogwarts school (which was founded at the end of the Xth century AC) might have been a bit different, or (unlikely) that the cultural exchange between the magic and the muggle community was extremely slow.

Although at the time Harry studies at Hogwarts there seems to be no discrimination whatsoever between students based on gender, still it seems that historically there are potions and spells accociated more with witches - the classic example is the brewing of various love potions (amorentia for example). It's intriguing that even through in the 1990s the knowledge or skills required for brewing the potions is not restricted to gender, the Weasley twins marketing efforts are still mainly focused on the witches, clearly they are the target audience as it is apparent from the labeling of the product and their sales speech :

“Haven’t you girls found our special WonderWitch products yet?” asked Fred. “Follow me, ladies…” "..." Near the window was an array of violently pink products around which a cluster of excited girls was giggling enthusiastically. "..." “There you go,” said Fred proudly. “Best range of love potions you’ll find anywhere.” Ginny raised an eyebrow skeptically. “Do they work?” she asked. “Certainly they work, for up to twenty-four hours at a time depending on the weight of the boy in question…” “…and the attractiveness of the girl,” said George. Draco's Detour - HBP

Tradition, I guess. There is also a wide range of cleaning and cooking spells used by our resident housewife, Mrs. Weasley. It may be that originally spells and potions like those also were considered 'witchcraft', being less sophisticated than the more 'wise' spells of Wizardry.

The founders, or maybe some later headmaster who gave Hogwarts its current name probably meant by it that the school teaches all kinds of magic, both 'witchcraft' and 'wizardry'.

Other than these examples, the distinctions, which once were probably more prominent, so much so that they required the creation and usage of two different words, ('witch' vs 'wizard') are almost nonexistent now in a more or less egalitarian magic community, and the words 'wizardry' and 'witchcraft' have the same meaning, (though unsurprisingly when referring to mixed gender groups the accepted term in our patriarchal language system is 'wizards' and not 'witches'.)

All this is pure speculation of course, it's equally possible that Rowling just liked the fancy name or that the name is about gender equality as @Gallifreyan suggests and the question @Valorum linked provides a sufficient answer and there is no difference between the two words other than 'witch' referring to female and 'wizard' to male individuals with magic ability, and we're once again overanalysing uncompicated matters and creating duplicates...