The Paired (with ratios or differences) approach pairs a control value with an experimental value i.e. C1 with E1, C2 with E2, etc. The Unpaired (or ANOVA) approach does no such pairing. The difference is that if samples really are paired, the variance in the denominator can be much smaller (and hence power greater). The paired t-test evaluates whether the mean difference between paired samples is significantly different from zero, whereas the non-paired t-test tests if the mean of group 1 is different from the mean of group 2. Let's do a "classic" example.
Below is the body weight of a group of people before and after some diet regime was applied:
PERSON BEFORE AFTER DIFFERENCE
Joe 200 180 -20
Sue 120 100 -20
George 175 155 -20
(Note: People here are like replicate experiments and Before and After are like Control versus Experimental). Doing a "paired t-test" results in showing the diet is very effective -- everyone loses 20 lb with no variance (that is: the mean difference is significantly different from zero). On the other hand, if we take the (Mean of the Afters) - (Mean of the Befores), the difference is still 20, however the variance within treatments is large so it is impossible to say the difference is significant.
Basically when observations are "paired" the paired approach is the correct and most powerful approach, when observations are not paired then we have to use the unpaired approach.