According to the DMG (187), NPCs in 4e seem to be run as a weird sort of template halfway between a monster and a PC - you don't generate an actual character with a full array of powers and feats. Is there a particular reason why I shouldn't create a fully-fledged character with PC levels to face off against the party, other than that this would make the encounter more challenging? The one issue I can see is that with more than one surge, depending on class the NPC's heals might make the battle drag on too long. Also, I can't seem to find the XP reward for an NPC as per the DMG - would it be the same as a standard monster of its level?
There are many reasons why NPCs built as PCs work differently than those built as monsters. 4e is carefully balanced and designed to make encounters interesting, and PC-class NPCs disrupt this design. Some reasons:
If you want to build an NPC like a PC despite all of this, there's no game-breaking reason why not to. Just be aware of the tradeoffs.
PCs don't have nearly enough HP to deal with a group of PCs attacking them. A L2 rogue can one shot a L2 or even L3 PC of nearly any build.
L2 charger thief (vanguard weapon, Light Blade Expertise, Surprising Charge) can do 2d8+3d6+7 dmg (1d8+41 on a crit) with a to hit of +14 (assuming CA)
considering that the best L2 PC has an AC of 22 (Knight with +1 Plate and a heavy shield) and 42 HP (16 CON,toughness as their L1 feat, whatever they want as their L2). You're talking about A PC doing 1/2 of a monster's damage on average per round and killing them ouright on a crit. PCs are not designed to fighter other PCs. PC targets need a lot more HP than other PCs have.
The thief is doing (9+10.5+7).60 + .05(41+4.5) = 15.9+2.28 = 18.75 means that your PC knight (optimized for AC not damage or movement or anything) lasts a bit more than 2 rounds. The average monster should stand up to about 4 rounds of average PC damage.
Consider that your L2 PC should be a significant enemy, you'd think he'd be a solo, or at least an Elite right? The only L2 solo I can find has 185 HP and an AC of 22. At best your L2 PC Knight is a brute, with about the right HP (a randomly pulled L2 brute had 43 HP, with an AC of just 14 (meaning that your Rogue takes him out in much less than 2 rounds). An Elite on the other hand has ~70 HP and defenses that are between the solo and the brute (~18 AC, high for L2).
Suffice to say, PC vs PC combat is a tricky business, a lot of PCs of various levels are perfectly capable of one-shotting the most durable PC of their level on a crit (the rogue takes out the Knight with a crit), and doing better than /2 damage on an average turn, these are odds you do not want your monsters/NPCs facing.
As noted in the comments, the converse of this concept is true as well. A well optimized thief (or other PC at higher levels) can one shot a party member. While this might be desirable on very rare occasions, it's not really a pleasant play experience.
Largely, it comes down to complexity and balance.
I want to start by arguing that you CAN effectively use full PC builds as NPC enemies. I ran a very effective and memorable encounter pitting 5 level 9 heroes against 4 level 10 PC enemies, who became memorable allies later on and one player took over one of them after his character meet a tragic and irreversible end.
That said, I spent over a month working on and fine-tuning the encounter. Each PC foe was fully built, with powers, feats, backgrounds, themes, and items. The group of four were painstakingly tuned for teamwork. I play-tested the battle several times to make sure the numbers matched up. I memorized each power for each player and built cheat-sheets with the hit and damage numbers for quick reference. It worked and my players were amazed, but it was tough to mentally keep track of 4 players whilst also adjudicating the rest of the fight.
I would, however, recommend against this in general. It's extremely demanding on the DM. Also, the encounter has to be very carefully tuned for this, because players fight in remarkably different ways than monsters. At-will damage is generally MUCH lower: level 10 defender doing 1d10+6 is nothing compared to a level 10 monster hitting for 2d10+10 or higher. The fight tends to last a lot longer too, especially if you have a leader PC enemy.
Ultimately, monster builds and NPC templates are designed as tools to facilitate your job as DM. You don't have to restrict yourself to those tools, but make sure you recognize the extra work involved in foregoing them. I would caution against randomly throwing a PC build in with other monsters without proper planning and expecting it to work as seamlessly as tossing in another monster, even a custom one.
I don't think there's any reason why you couldn't, but plenty of reasons why you shouldn't.
Ultimately it comes down to time; that is your time. It takes a while to build a PC, and if you fling that creature into combat with the party they're almost certainly going to lose. Depending on the level, they will probably have failed to use most of their encounter powers and probably most of their daily powers too during that combat.
Instead of building a PC for them to fight, you'd be better off thinking of ways to make the fight end in something other than a one-sided defeat, and having the resulting NPC be beneficial in the future.
If you are building the NPC to be versatile over several encounters, then I think you'd be better off just planning out the NPC for each combat as it comes, with abilities as appropriate; it's entirely possible that later combats might not even take place.
To address one of your points, rebuilding a monster as a PC won't make it more dangerous - monsters are mostly limited by their actions (one per average five of the PCs), not the variety of their abilities. If the whole encounter is "PC" monsters, you won't be able to run it very well since you'll forget most of the synergies and abilities and just end up doing basic melee attacks or similar anyway.
My suggestion, think about how the NPC is supposed to act, and build one or two powers that exemplify this nature.