What are the advantages of AD&D over D&D3?

This question is very much about personal preference; therefore there isn't going to be a "right answer" or "solution." Here are my reasons for still playing:

1. AD&D 1e is the last edition where a player's skill during the game mattered more than their skill during the character creation process (if we ignore some of the changes introduced by Unearthed Arcana, which I do). Standard character creation methods leave no room to min-max or optimize character builds.
2. AD&D is a far more modular game than more recent D&D iterations -- bits and pieces of it can be hacked or removed (and completely new bits inserted) without upsetting the system's balance. Some would argue that this simply reflects a lack of balance; I think it's because the core experience of the game has nothing to do with many of these systems.[1]
3. AD&D often plays better not as a combat-centric game, but as an exploration-centric game in which the outcome of combat must occasionally be adjudicated. This explains the XP system, the rigorous systematization of movement outside of combat (rather than inside), the resource management/encumbrance rules, and perhaps even the organization of the rule-books themselves.
4. AD&D does not make the player feel heroic by default; it is a bloody and desperate affair until the party earns their money/power. It also doesn't pre-judge what activities should be the domain of adventurers -- anything that makes a buck is fair game (except for paladins).

Of course, there's plenty more that could be said -- and other people will have completely different lists and/or look at this list and only see evidence that 3E is superior. Like I said at the beginning, there are no right answers.

[1] OD&D has this too, of course, but AD&D comes pre-packaged with systems (e.g., psionics, grappling) just begging to be hacked.

There are three reasons I can immediately think of that apply to new players - they're the reasons that my group started with an earlier edition of D&D, rather than later ones. Early D&D has much simpler and faster combat, with most attacks just being a roll of 1d20, a table lookup, and a damage roll if successful. Spells require a save or attack roll, and maybe a damage roll. For new groups or large groups, it flows easier. Also, new DMs don't need to learn as much to have simple combat. Mostly at higher levels, WotC D&D takes a lot more time, especially for player decision-making, and the higher complexity also requires more system mastery of the DM and players.

D&D 3.x also has many more classes, spells and other options with strange interactions. First edition didn't have as much focus on RAW, so a rules-based exploit would be quickly shut down by a DM, even a new one, if it didn't make sense in-world. But 3.x has a fair number of Pun-Pun possibilities, which are often hard for a new DM to avoid easily, especially since (to a novice) they don't always seem so dangerous until after it's too late.

Lastly, when my group started, we wanted to play without miniatures. Nowadays we use them for large battles, but they aren't ever necessary and most fights go just as smoothly, and even a little quicker, without them. The lack of necessary investment makes it easier to play AD&D out of the box.

None of this is necessarily true for every group, but it's what worked for us, and I think in our case we made the right choice.

A lot of the time, though, people simply started with an earlier edition of D&D and don't want to move on. It sounds like you might not like D&D 4e, or 3.5e, since you don't mention playing them, or simply didn't bother trying them because you were happy with the edition you were already using.

Lastly, some of the things you may like about 3rd edition (and some of the 'flaws' in AD&D) might be viewed differently by different groups. I personally don't like the encounter xp budget systems in later editions, even though other people really like them, because it makes the world fit the party rather than the party adjust to the world. I like my group to judge threats and decide whether to fight or run, rather than just operating under the assumption that everything they meet is something they can kill. Similarly, other people don't like how long it takes to heal back HP, while I think having consequences last for a week rather than until the next rest makes combat more interesting, as the stakes are higher.

In the end, it's a matter of personal opinion for a lot of people.

The reasons I would give are vary similar to Craddokes answer. I will comment that most gamers don't play AD&D by the book. In my experience the most common variation is the AD&D character options, items, spells, and creature combined with OD&D/BECMI style combat leavened with a handful of rules from the DMG.

This is due what I considered the poor presentation of much of the combat system in the DMG. You can read one detailed interpretation of the by the book rules in the ADDICT Document.

However this variation has proven very popular and allowed AD&D to endure in popularity to this day.

The OSRIC, Adventure Dark and Deep or Advanced Edition Companion, are some example of cleaned up retro clones of AD&D first edition.

I will advise anybody using AD&D first edition for the first time to read Matt Finch's Old School Primer. It very helpful in understanding how to apply the rules light older editions in a campaign.

And for those interested, like all RPGs AD&D abstracts combat, and characters to make a playable game. What many don't realizes that AD&D is a result of several years of development starting with Dave Arneson's original Blackmoor campaign. This development is why AD&D has the mechanics it does.

When Dave Arneson wanted to run a Welsey style Braustein game of his own he opted to set it in a medieval castle and town called Blackmoor. The starting point for his rules was the Chainmail miniature wargame.

Chainmail had a fantasy supplement that not only allowed for fantastic monster to be incorporated but hero types (or anti-heroes as the case may be).

A Hero was worth four normal chainmail figures and took four hits to kill. A super hero was worth eight figures and took eight hits to kill. The one hit = one kill system proved too deadly so a d6 roll was substituted for damage. And hits to kill became 1d6 hit points.

In addition intermediate points were added to character progression. In the development of the Blackmoor and later the Greyhawk campaigns, characters no long jumped form being a veteran to a hero. But went through a progression where they jumped to being worth two veterans, then three, then finally they became a hero worth four veteran warriors.

The same system was applied to magic users. Later when the players of Blackmoor found it difficult to handle another player who became a vampire, David Fang, the cleric class was created. A combination of Van Helsing and Charlemagne's Bishop Turpin.

Knowing this helps a gamer realize that AD&D isn't just a random assemblage of mechanics but rather was grown through numerous play test campaigns in response to what the players were doing. Even when this is understood a gamer still may opt to pick another system that better suited to his sensibilities.