Caveat: Note that none of the answers on this question are the correct answer, as the rules as written are ambiguous and do not address this. Rather, the answers here provide different possible ways to resolve the ambiguity. Some of the answers, mine included, do not break the rules as written, do not require any house rules, and are different ways of interpreting the rules as they are written. While we can argue about which of these types of answers fits the rules best logically, I think that this answer is the one which provides the most fair results out of all the different ways the ambiguity could be resolved by the rules as written.
One answer to the question about "What is the difference, if any, between permanent and until dispelled?" has a comment, by @FooBar, which says:
Dispel Magic says "For each spell of 4th level or higher on the
target, make an ability check using your spellcasting ability." There
are 365 castings of Teleportation Circle on the object. Once you make
365 successful dispel checks, the circle is destroyed. This will
probably take less than 365 dispel castings, but more than just 1.
I have encouraged that user to turn their comment into an answer.
I think this is a great way to view the permanent circle. It's not that "the last casting becomes permanent", which did not make sense to me anyway. It's not that the first is permanent, or any other single casting. They all become permanent together, as they are all working together. This seems more logical and would fit the description better.
By the time you get to the very last day and make the final casting, there must still be something left of all the previous castings. Some magical energy, spell residue, maybe just the universe remembering what you've done, but something. Otherwise, on that last day the last casting would behave just as the first, and indeed the spell would "think" that it was the first. Each and every casting, down to the 365th, would be "thinking" (for lack of a better way to put it) "Yay! I'm a new circle! And I'm the first one. I hope my caster makes 364 more of me so I become permanent."
Indeed, if I may borrow from my computer background for a moment, if I were to write a computer simulation of a D&D world and included this effect in the simulation, I would have to keep the previous 364 spells around in some way, shape, or form. If I did not have anything at all to keep them around or remember them, then there would be a bug in the simulation and the last casting (and all others) would think that it was the first and no permanent circle would be created.
So if something persists, what is it exactly? Unfortunately for my answer, it is not the full spell effect as you might think of it. It cannot be the full spell effect still hanging around at full power, or the previous circle would still be usable. So it must be some secondary effect of the spell which has a separate duration which is a custom, special duration.
"Wait, but the rules don't say it is a secondary effect with a separate duration!" Yes, that is true because the rules are ambiguous here, so we have to resolve it somehow. And this seems one reasonable way to view the effect.
So something remains. And that something must be building up more and more power over the year. We cannot just say that each special duration ends when the next one begins and remembers nothing from before, otherwise we are back to a similar problem again, as the spell doesn't "know" how many came before.
You can look at this next part in multiple different ways, but they are all essentially equivalent to each other. Either the secondary special duration of each casting lasts until the final and complete ending (365 day completion, or until it "fizzles" from failure to finish), or equivalently there is 1 effect that just keeps growing and gaining more and more power over time.
Saying it is 1 effect with more and more power is fine, except that it leaves you wondering "If it's more powerful than a spell of that spell slot, then how difficult is it to dispel?" and we're right back to where this whole question originated from, and you just have to house rule an ad-hoc modifier to a dispel check.
But if you look at it the other equivalent way, saying that each effect lasts until the end, then you no longer have to house rule the thing which is definitely stronger, and instead you have a straightforward way to let the rules handle themselves. If each effect is still present separately instead of as 1 overcharged effect, then the rules suggest you do a dispel check against each.
How this affects the game
If you look at it this way, you can easily justify allowing the circle to be dispelled and at the same time make it harder to dispel in a way that conforms to the rest of the raw rules. So it is beneficial from a balance perspective and also from a fairness perspective.
Note that this does not require you to dispel once per day for a year, but rather just 365 (dispel checks, not castings) times whenever. So you could succeed at 10 checks per day for just over a month, or more to do it all in 1 day. Or random people could try once or twice to dispel it every now and then, and it could slowly add up to be totally dispelled in a decade or a century.
Unfortunately, this reading does provide the worst possible book-keeping requirement out of all the answers. You would need to keep track of each and every single casting which went into the circle initially, and then you would also need to keep track separately of each and every successful dispel.
Silver lining: This means it would indeed be beneficial to get a 9th level casting in there every now and then to deter low level dispellers from slowly whittling away at it. They could easily dispel the effects one at a time until they got to the next 9th level one which would slow them down for a while.
I don't think the above silver lining is worth all the book keeping that would be necessary, so I would not go with it exactly as written above, but such deviations I made would be house rules.
Despite the annoying book keeping, this strikes a good, rule-abiding balance between "I should be able to dispel that somehow; you can't just say it's invincible" and "You are going to say the villain can ruin my year of work with 1 dispel?!?!?11one Well ---- you all, I rage-quit your stupid game."
Some side effects of this reading
If it takes the power of 365 concurrent instances of the effect to be permanent, then even if you are correct then dispelling only 1 of them leaves you with 364 concurrent instances of the effect, which makes it equivalent to where the progress was the day before it was finished. So does it fail anyway?
This then becomes another ambiguity. If you don't want to make house rules for this, you could treat it the same way you treated the original "long-cast" on the 364th day, since the power would be essentially equivalent to that. So the effect would stop working, yes, but you would still have the potential to save it if you recast the spell to complete the 365 instances again before the end of the following day.
Same as before, but what if 10 successful dispel checks were made against it?
Then the answer is likewise similar. It is now equivalent to when you were 355 days in, so you need to start casting once per day again for the next 10 days.
Just like this entire Q&A in general, these side-effect questions result from an ambiguity, so you could rule differently. I think these responses to the side effects are the way to most closely adhere to the rules and introduce no house rules. And it adds interesting flavor and mechanics and still seems more fair and balanced than the entire effect simply failing completely.