How can a paired test less powerful than unpaired test? I'm using wilcox.test to compare two timepoints as such:
a <- c(1,2,3,4,5)
b <- c(2,3,4,5,6)
wilcox.test(a, b)

The result of the rank sum test is, as expected p > 0.05 (0.3976)
If I treat the samples as paired, the result is, as expected more significant: 
wilcox.test(a, b, paired=T)

with the signed rank test coming up with a p-value of 0.007937.
But I don't understand why the same doesn't hold true with actual data from my experiment:
a <- c(244.5, 242.3, 225.25, 250.15, 254.0)
b <- c(196.8, 186.25, 175.75, 174.75, 170.33)
wilcox.test(a, b)

Rank sum test: W = 25, p-value = 0.007937
wilcox.test(a,b, paired=T)

Signed rank test: V = 15, p-value = 0.0625
I don't quite understand why the latter is less powerful in this example. Is there a better test to use to test the hypothesis that the measurements differ between these two timepoints (without assuming a normal distribution)? 
 A: Paired tests leverage the correlation between your variables (typically before and after) to improve power.  The more strongly, and positively, your data are correlated, the more of the variability that exists between the units (e.g., patients), and which gets ignored by the test (although if that variability would have otherwise supported the alternative hypothesis, this looses power, see comments below).  If the data were uncorrelated, there shouldn't be any long run gain, and if the data were negatively correlated, taking that fact into account (which is the correct thing to do) shows that there is actually less information in your data than might naively appear.  Let's look at your data:  
a <- c(1,2,3,4,5)
b <- c(2,3,4,5,6)
cor(a, b)
# [1] 1
wilcox.test(a,b)
#  Wilcoxon rank sum test with continuity correction
# 
# data:  a and b
# W = 8, p-value = 0.3976
# alternative hypothesis: true location shift is not equal to 0
wilcox.test(a,b, paired=T)
#  Wilcoxon signed rank test with continuity correction
# 
# data:  a and b
# V = 0, p-value = 0.03689
# alternative hypothesis: true location shift is not equal to 0


a <- c(244.5, 242.3, 225.25, 250.15, 254.0)
b <- c(196.8, 186.25, 175.75, 174.75, 170.33)
cor(a, b)
# [1] -0.1026423
wilcox.test(a, b)
#  Wilcoxon rank sum test
# 
# data:  a and b
# W = 25, p-value = 0.007937
# alternative hypothesis: true location shift is not equal to 0
wilcox.test(a, b, paired=T)
#  Wilcoxon signed rank test
# 
# data:  a and b
# V = 15, p-value = 0.0625
# alternative hypothesis: true location shift is not equal to 0

