Buffy takes place in a world that existed before social media - as the show ended in 2003, at a time when social media was still in its infancy. For the years that Buffy and the Scoobies were in high school, very little was digital, and Buffy had... a pager, for those moments when Giles needed to get hold of her so that she could save the world.
Later in the series, some level of technology did appear with the Initiative, and Buffy made it clear that she actually preferred not to use tech to track the vamps she hunted, or to try and take them down. Turns out, Slayer-powers and skill are worth more than all the fancy weaponry and tracking tech she could access. So it's unlikely she'd be making a major switch in her actual Slayer duties, but if social media, smartphones, and the internet of today existed then, some other things definitely would change.
Cordelia Would Be A Teen Influencer
Cordelia was always the queen bee of Sunnydale High (at least, until she started to develop a closer relationship with the deeply unpopular Scoobies). But in the age of social media, she would probably be popular online, too! Between her parent's money (and the fact that she always had the latest fashions and a fancy car), and her own charisma, Cordelia would no doubt be an influencer, even as a teenager.
It Would Be Harder To Convince People Vamps Were Just Criminals
Throughout the early seasons in particular, vampire attacks were often explained away by people decided that the vampires were actually just violent criminals, with people believing that the vampiric ridges on their faces were the result of self-multilation or some strange kind of drug. Other supernatural attacks were also often explained away on a larger scale - like the destruction of the school, which was explained as some kind of accidental explosion (rather than giant-demon-snake). However, with smartphones and everyone recording attacks, it would be a lot harder to keep the Hellmouth a secret.
Giles Would Hate It... But Learn To Work With It
Even in the '90s, Giles hated the technology that existed, and was upset at computers in his library and having to scan books in, as well as Jenny Calendar's love of all things 'techno pagan'. However, he would have had a field day dunking on social media.
Giles would consider it shallow, superficial, and ridiculous, but over time, he would probably be won over by how useful it turned out to be in finding potentials and staying in contact with people to know they are safe.
And Willow Would Hate The Social Side, But Love The Tech
Willow was never a character who was overly popular, but she was already the tech genius of the scoobies by the end of the series, and this would just be even more pronounced if she had access to social media. Willow would be able to find out where anyone was, at any moment, and would probably take her hacking skills (practical and magical) to a whole new level... but would still struggle to gain her own following, at least as a teen.
Willow's Malcolm Storyline Would Change
In the original series, one of the biggest tech-related storylines was when Willow ended up falling for a demon, who had been accidentally transferred into the school's network. A lot of this would still work, but a few things might have to change - for one thing, while it's totally normal to meet people online now, Willow would expect to see photos, and a social media presence and Moloch might be found out to be a demon a whole lot faster... and might find it harder to corrupt people, if he had to vie for their attention in the social media world!
Anya Would Be Obsessed With Followers
Anya, after becoming human, quickly became obsessed with money - and with all the ways that she could make it at the Magic Box. Similarly, she would no doubt become obsessed with the social media for the store, and with converting followers into customers, and getting those followers in the first place. Of course, the rest of the gang might have to calm her down a bit, too, especially if she got a negative review.
Willow & Tara Would Have An Online Witch Group
In the original series, Willow and Tara met at a witch/pagan group at university, where they were the only two members who really seemed interested in witchcraft, rather than an 'empowering' bake sale.
In the modern day, of course, they would end up dealing with similar frustrations with online witch 'aesthetic', but would be able to connect with witches all over the world in groups and communities.
Buffy Would Be Able To Make A Living Online
After Buffy comes back from the dead (the second time), she has to get a job at a fast-food place to make ends meet. She needs money to support Dawn, and as a dead person with no work experience, she's incredibly limited in Sunnydale when it comes to employment opportunities. In the modern social media age, though, she would be able to make a lot more money, with a far more flexible schedule, by finding a way to use her skills online. A YouTube self-defense channel mixing her workouts and training with her bubbly personality? The perfect way for a Slayer to succeed.
There Would Be At Least One Supernatural-Social-Media Story
While there was Malcolm in the original series, and a few hints at technology and Slaying combined, there would be significantly more in a social media world. Especially in the high school years, when storylines were very clearly focused on 'normal' teen issues with a supernatural twist (like mean girls, feeling invisible, pressure from athletic coaches, or cheerleading tryouts), social media supernatural episodes would be the norm.
No More "If The Apocalypse Comes, Beep Me"
One of the best-loved lines in Buffy is when she quips to Giles 'If the apocalypse comes, beep me!'. The issue is that Buffy wants to go on a date, and Giles wants her to stay put in case something happens and they need her to be the Slayer. However, in a world of smartphones, this would be totally unnecessary. Buffy would just have a phone, so would Giles, and keeping her chained to the library would be completely pointless. We'd miss that line, though.