This depends on your campaign/DM
To be honest, this seems like a question that can all too easily become purely opinion based, so I will try to focus only on my experience in playing a Fairy (Specifically the Fairy race from Mordenkainen Presents: Monsters of the Multiverse), and how it felt playing a fae alongside mostly humanoid companions.
As a preface: the campaign I play a Fairy in is set in a homebrewed setting, and as such the DM has and does take my Fairy features into account when planning encounters and such.
Generally, there's no balance difference, but flight can be dangerous
On the whole, outside of debating whether the spell immunities/vulnerabilities play into it, as NotArch has so aptly put it all officially published races tend to be balanced, or at least not so unbalanced that you will feel obviously behind any other player for playing a certain race.
However, something I feel that may have been ignored is how the Fairy race has access to a fly speed. A 30ft. fly speed can certainly make some challenges trivial for you where your other players may struggle. This is especially true in low level campaigns, and/or settings with few magic items.
As an example, in the early stages of my campaign the party came across a fast flowing river, 60ft. across, with a key on the other side we needed to progress. Given we were only 2nd level, without my character's ability to fly this would have certainly been a very challenging feat.
Of course, there are non-fae races which have a fly speed, but these are already considered quite powerful. Just a grain of salt to take with playing a Fairy specifically, as you seem intent on.
The spell changes balance out, so long as your DM isn't cruel
While it's true that some PC features can cause you some grief as a fey (Ancients Paladin's Turn the Faithless comes to mind), from my experience there are so few spells that actually impact you that the chances of encountering an NPC with access so any of those spells is minimal. This, of course, varies by setting, and a particularly cruel DM can simply throw a ton of fey-hunting rangers or other such threat at you if they want to make a point of your character being non-humanoid.
In my campaign, we have fought against hostile mages on occasion, some of which carried hold person, but since I was playing a high-wisdom character anyway it was unlikely I would have been the target of those spells regardless, and then even if I were targeted I had a high chance of succeeding the save, so being immune to the spell did not have a huge impact. We have also fought against spellcasters with protection from evil and good, but in those cases this just meant I focused my efforts on other aspects of the fight, like attacking other creatures, or in my case specifically healing/supporting the other party members. Thus, on both sides of the spell differences I found it really didn't impact my power level in combat. I can't speak too much for out of combat, however.
Overall, I would say there's no real balance issue here, excepting of course the fly speed, which is something you should check with your DM about anyway (A thoroughly overused saying on this SE, as I understand, but nonetheless the best advice here). Happy Fairy-ing!
