Q-Q plot interpretation Consider the following code and output:
  par(mfrow=c(3,2))
  # generate random data from weibull distribution
  x = rweibull(20, 8, 2)
  # Quantile-Quantile Plot for different distributions
  qqPlot(x, "log-normal")
  qqPlot(x, "normal")
  qqPlot(x, "exponential", DB = TRUE)
  qqPlot(x, "cauchy")
  qqPlot(x, "weibull")
  qqPlot(x, "logistic")


It seems that that Q-Q plot for log-normal is almost the same as the Q-Q plot for weibull. How can we distinguish them? Also if the points are within the region defined by the two outer black lines, does that indicate that they follow the specified distribution?
 A: 
It seems that that Q-Q plot for log-normal is almost the same as the Q-Q plot for weibull. 

Yes.

How can we distinguish them? 

At that sample size, you likely can't.

Also if the points are within the region defined by the two outer black lines, does that indicate that they follow the specified distribution?

No. It only indicates that you can't tell the distribution of the data as being different from that distribution. It's lack of evidence of a difference, not evidence of a lack of difference. 
You can be almost certain that the data are from a distribution that is not any of the ones you have considered (why would it be exactly from any of those?). 
A: There are a couple of things to be said here:  


*

*the shape of the CDF for the log-normal is similar enough to the shape of the CDF of the Weibull to make them harder to distinguish than the level of similarity between the Weibull and the others.  

*the outer black lines form a confidence band.  The use of the confidence band in inference is the same as any other standard form of Frequentist statistical inference.  That is, when values fall within the band, we cannot reject the null hypothesis that the posited distribution is the correct one.  This is not the same as saying that we know the posited distribution is the correct one.  (Note that this is a great example of what I discussed in another answer here of a situation where the Fisherian perspective on hypothesis testing would be preferable to the Neyman-Pearson.)  

*you need more data; your $N$ is only 20 here.  

