Bardic Magic vs Magician Magic
So, bardic magic vs the magic we see in Harry Potter. What are the differences? Well, let's break them down.
Magician
So, how does Harry Potter magic work exactly? Well, there's the use of a focus, verbal components, and somatic components. The wand, the incantation, and the wand movement. We know that magical strength is a measurable phenomenon, seeing as Dumbledore said
"Voldemort will not have cared about the weight, but about the amount of magical power that crossed his lake. I rather think an enchantment will have been placed upon this boat so that only one wizard at a time will be able to sail in it."
in The Half Blood Prince. There are clearly rules to how the magic works, both about the limitations of magic, and the importance of proper pronunciation of the verbal components, seeing as Flitwick says
"[...]And saying the magic words properly is very important, too — never forget Wizard Baruffio, who said ‘s’ instead of ‘f and found himself on the floor with a buffalo on his chest.”
as well as the reference to Gamp's Law of Elemental Transfiguration, of which there are five exceptions. The only known one is that food can't be conjured out of nothing, implying that the other four are based on conjuration as well. There are also the Fundamental Laws of Magic from Adalbert Waffling, of which we only know the first: The further one delves into meddling with the underlying laws of magic, the greater the consequences. Then there's Golpalott's Third Law, which centers on antidotes to poisons, namely that to make the antidote for a blended poison, you had to find the antidote to each poison as well as the ingredient that would blend the antidote. There's also the Principle of Artificianimate Quasi-Dominance, which is used to explain why things can go wrong when conjuring living creatures, such as a severed head. Professor Croaker's Law centers on five hours being the maximum time one can go back without possible harm being inflicted on the traveler or the time stream.
The lack of a focus or training makes it so that magic is dependent on need and emotion, occurring through accident. The study of magic itself is separated into various branches, such as Potions, Alchemy, Herbology, Wandless Magic, etc. Foci are subject to their branch of magic, namely Wandlore and the Subtle Laws of Wands. These laws dictate things like wand bonding, allegiance, the ability to win allegiance, and the magic branches that they tend toward.
So Harry Potter magic is regulated by laws, formulaic, and teachable. Unpredictable results arise from failure to properly adhere to the formula, or no effect is created. A focus is required for magic to be regulated, however advanced practitioners are capable of performing specific spells without a wand.
The ability to use magic is, at least partly, genetic. Magical parents have magical children. Word of God established that Muggleborns have Squib ancestors somewhere in their family tree. Squibs are children of magical parents who cannot use magic.
Changes
First of all, I hate the whole "squib lineage" thing, it's dumb and the idea that magic is tied to bloodlines just validates the blood purism in the books in a way, because "Oh, the muggleborns aren't actually from Muggles, they have magical heritage!" You see the issue? So, getting rid of that, and it'll never be a part of the worldbuilding for any Harry Potter fanfic I ever write. Instead, we'll be going the route I went with the Monster Club AU. The ability to use magic is genetic but in a different way. There are a set of genes that determine magical things, the ability to use it, the ability to perceive it, metamorphmagus or not, that stuff. The majority state for these is dormant, which is why most people aren't magical. People who are magical have the active version of the gene, allowing them to use magic, to see it, and other things. It's a random chance, however, which is why non-magical people can have magical children. While being magical increases the likelihood of your child having the active variant of the gene, they can still have the dormant gene, leading to Squibs.
The Use gene, Gene U, is what allows magical people, human or not, to tap into something called "The Wellspring", the global field of ambient magic, with the amount you can draw in being up to chance, but slowly growing as you age. The active gene then allows other genes which may be active to express themselves. As an example, metamorphmagi. Someone with a dormant Gene U may have an active Gene S (self-shift), but the lack of access to The Wellspring means it can't express, but an active Gene U allows Gene S to express. Gene U is a separate gene from Gene P, which is what lets people perceive magical things. Squibs have a dormant Gene U, but an active Gene P, which is why Filch can actively see Hogwarts (which looks like a ruined castle to non-magicals) but can't actively use magic.
Gene P tends to be active in parents of Muggleborn magicians, as well as being generally more active in a greater amount of the non-magical population, leading to things like having a "sixth-sense" and ghost sightings. The people who can't see ghosts, or can't feel when they're being watched, or anything along those lines, have a dormant Gene P. Gene P being separate from Gene U isn't just a way to logic out how Squibs work, it's also the reason that people can interact with magic in unique ways, which brings us to:
Bardic
So, how is bardic magic different? Well, it's predicated on two things. The first, a gene we'll call Gene R, which affects Gene P, the second, is essentially a mutation of Gene U. Gene P allows perception of magic, but Gene R expresses itself as hearing magic, and natural rhythms. The mutation of Gene U allows manipulation of magic through different methods, such as music, in a bard's case. Without Gene R, the magician would go about like a regular magician does, just using a wand, incantations, and the like. With Gene R they can hear a different method that, if trained, will become the means by which they use magic. Of course, it's possible to have the mutation without Gene R, which would make it so that you wouldn't be able to use bardic magic, but could use other systems, such as using spinning wheels, looms, and metaphysical thread to weave spells together.
That's the why though, not the how. In short, bardic magic relies on music, nature, and State. If something is fully manmade, then it can't be done with bardic magic. We'll use Alohomora as an example. Wanded magic, predicated on a door being locked, used to unlock the door. A bard wouldn't be able to make an equivalent, because the spell is contingent on a fully manmade State. Locked, as a State, doesn't exist outside of the bounds of the manmade object of locks, so it isn't a State that could be altered by song. Rust? A natural State, thus one that can be altered by song. One of the things that bardic magic is incapable of effecting is plastic. And, generally, anything wholly manmade, so stuff like nylon would also be precluded from being affected. The logic is that they aren't "natural" on a metaphysical level, even if they can possibly exist naturally, under hyper-specific circumstances. Certain effects can only be made by using different types of instruments, which are dependent on how the bard maps the rhythms they hear to the instruments and music they know about like percussive elements being needed for subtle defensive magic.
Additionally, talismans are something that is an aspect of bardic magic, where singing the right songs at the right items, or using the right reagents enchants the entirety with the intended effect. While talismans are enchanted, there is a bit of a metaphysical distinction I'm drawing between a talisman and an enchanted object. Talismans provide the wearer a specific capability or boost, whereas enchanted objects don't. Alongside that, talismans have their effect as an innate aspect of how they exist, whereas enchanted objects have the enchantment... sort of draped over the item. Harry's arrowhead necklace, for example, is a talisman, whereas a fireplace with a fire that doesn't require fuel would be an enchanted object. The former allows him to never stray from his intended goal, the latter simply has an enchantment lying overtop of it.