If you actually get a standard distribution from the dice in the 3d6 x12 method, it will be slightly better than a standard distribution of results from the 4d6 method. The more samples you take, the more likely it is that you will get something approaching average or a standard distribution. The fewer samples you take, the more likely the results will just be random.
A good way to analyze the differences between the two distributions is to imagine a head-to-head contest between characters.
First, suppose you have two identical characters, A and B, rolling off against each other with d20. They tie 5% of the time; 47.5% of the time one wins; 47.5% of the time the other wins. In contrast, if you use 3d6, ties occur 9.2% of the time and each wins 45.4% of the time. Not a huge deal. Let's discard the ties and just concentrate on who wins more, A or B. Now let's start giving them bonuses. Since we haven't said who is whom, we'll just declare that A is the stronger one and B is the weaker one.
A's bonus 3d6 d20 3d6 ratio
========= ===================== ===================== over
========= A-wins B-wins ratio A-wins B-wins ratio d20 ratio
--------- ------ ------ ----- ------ ------ ----- ---------
+0 45.36% 45.36% 1.0 47.50% 47.50% 1.0 1.0
+1 54.64% 36.31% 1.5 52.50% 42.75% 1.2 1.2
+2 63.69% 27.94% 2.3 57.25% 38.25% 1.5 1.5
+3 72.06% 20.58% 3.5 61.75% 34.00% 1.8 1.9
+4 79.42% 14.46% 5.5 66.00% 30.00% 2.2 2.5
+5 85.54% 9.65% 8.9 70.00% 26.25% 2.7 3.3
+6 90.35% 6.08% 14.9 73.75% 22.75% 3.2 4.6
+7 93.92% 3.59% 26.2 77.25% 19.50% 4.0 6.6
+8 96.41% 1.97% 49.0 80.50% 16.50% 4.9 10.0
+9 98.03% 0.99% 99.0 83.50% 13.75% 6.1 16.3
+10 99.01% 0.45% 220.0 86.25% 11.25% 7.7 28.7
+11 99.55% 0.18% 552.9 88.75% 9.00% 9.9 56.1
+12 99.82% 0.06% 1663 91.00% 7.00% 13.0 127.9
+13 99.94% 0.02% 6661 93.00% 5.25% 17.7 376.0
+14 99.98% 0.00% 46649 94.75% 3.75% 25.3 1846.3
Okay, so what does this tell us?
First, we can see that with big bonuses, A slaughters B head-to-head in rolls in 3d6, whereas with d20 the benefit that A gets over B is pretty modest (has to get all the way up to +11 before A is tenfold more likely to win than B!).
But, second, if you look at the ratio of ratios (that is, how much advantage A vs B has in 3d6 compared to A vs B in d20), you find that in 3d6 the bonus is pretty much squared compared to d20 (low values only--then it gets way, way more extreme later on).
So, what does this mean? Well, basically, if under 3d6 you have a +1 bonus more than someone else, it feels like a +2 difference in d20. +7 feels like +14.
So the concise explanation is: moving from d20 to 3d6 amplifies differences, making them feel about twice as large as before. (Of course, almost nothing is actually resolved as a head-to-head test, but it's a useful thought experiment.) You can cleave through hordes of lesser beings with that much more ease, and your betters become that much more fearsome. In fact, better just stay away from them. There are some kobolds that need slaying. Right? Right.
Best Answer
No, It's Not
You can use AnyDice to visualize dice rolls really easily and see what's going on. The links will show a table with the results for each one.
Here's 3d6.
Here's 1d18.
Here's both on the same screen for easy comparison.
Aside from the obvious issue of not being able to roll 1 on 3d6 because the minimum on each die is 1, the numbers in the middle will come up more often because there are more combinations that make them occur.
You're right, 1+1+1 and 6+6+6 are equally common, but there are 6^3 (216) possible permutations of results in 3d6 and only one of them comes out to 18. Quite a few of them come out to 9 though (6+1+2, 3+3+3, 4+2+3, etc). Comparatively on 1d18, there are 18 permutations and one of them is 18. Every number is equally likely. Thus, rolling 1d18 is going to be a lot more "swingy" (seeing the highs and the lows more often) than 3d6 will be. This has implications if you're making a dice rolling system, particularly if you have some kind of critical success or fail at the extremes. You'll see a minimum roll on 1d18 far more often than you'll see a minimum roll on 3d6.
Here's a good article explaining the math behind it. Any questions beyond that are probably better asked on Mathematics.SE. :)