Let me suggest
www.anydice.com. It will do the probabilities of most dice rolls for you. He's the results for your question.
The table below lists the probability that the roll will add up to at least the number value on the left. So:
"rolling 4d6 once",
#,%
3,100
4,99.9228395061728
5,99.6141975308638
6,98.8425925925918
7,97.2222222222218
8,94.2901234567918
9,89.5061728395118
10,82.4845679012418
11,73.0709876543318
12,61.6512345679318
13,48.7654320988318
14,35.4938271605318
15,23.1481481482318
16,13.0401234568318
17,5.7870370370818005
18,1.6203703704118002
"rolling 3d6 twice and taking the best roll",
#,%
3,100
4,99.99785665294925
5,99.96570644718796
6,99.78566529492495
7,99.14266117969895
8,97.37439986282895
9,93.27846364883895
10,85.93750000000895
11,75.00000000000895
12,60.93750000000895
13,45.13031550070895
14,29.78180727020895
15,17.66117969820895
16,9.044924554178952
17,3.669410150888951
18,0.9237825788689511
So as you can see, you end up with about a 2% higher chance of rolling above 11 by rolling 4d6 than 3d6 twice. Don't know if that is enough of an effect for you. Here's the setup I used. Try it yourself and play around until you get the results you want:
output [highest 3 of 4d6] named "rolling 4d6 once"
output [highest of 3d6 and 3d6] named "rolling 3d6 twice and taking the best roll"