When looking at the association between binary variables, when are odds ratios better than risk ratios and vice-versa? The Wikipedia page lists some scenarios:

While both measures are useful, they have different statistical uses.
  In medical research, the odds ratio is commonly used for case-control
  studies, as odds, but not probabilities, are usually estimated.
  Relative risk is commonly used in randomized controlled trials and
  cohort studies. When the incidence of outcomes are rare in the study
  population (generally interpreted to mean less than 10%), the odds
  ratio is considered a good estimate of the risk ratio. However, as
  outcomes become more common, the odds ratio and risk ratio diverge,
  with the odds ratio overestimating or underestimating the risk ratio
  when the estimates are greater than or less than 1, respectively. When
  estimates of the incidence of outcomes are available, methods exist to
  convert odds ratios to risk ratios.

However, it does not explain why risk ratios are used some contexts and odds ratios in others.
 A: The reasons for using odds instead of relative risk are probably easier to see if the formulas are first put on the table.
Relative Risk =
$\frac{\mbox{Incidence in exposed}}{\mbox{Incidence in unexposed}}$
Odd Ratio = 
$\frac{\mbox{Odds that an exposed person develops the outcome}}{\mbox{Odds that an unexposed person develops the outcome}}$
The main difference between the two ratios appears to be that the RR uses actual incidence rates, and the OR uses odds. So we might ask when odds would be obtainable and the actual incidence would not. The answer, though not explained, is alluded to in the quote you provided. By this I am referring to the point that ORs are used in case control studies, and RR is commonly used in randomized controlled trials (RCT) and cohort studies. 
Recall that incidence is a measurement of the number of new individuals who contract a disease during a particular period of time.
In a cohort study, the researchers can identify incidences. They first identify the presence or absence of a risk factor (say smoking) in two groups. Then they follow those groups for the length of the study. In the end, some members of both groups have probably experienced the outcome of interest and some have not. The incidences in the numerator and denominator can be calculated and the ratio obtained. An RCT yields similar information and can therefore also be used to calculate an RR.
In a case control study, the researchers identify a group of subjects who have the outcome of interest (group A), like lung cancer, and a group that does not (group B), but are otherwise similar to the people in A. The investigators then determine in some way (say interviews) if members of A and B have been exposed to a risk factor (like smoking). Now, speaking to your question -- the researchers cannot calculate incidence (risk) in the two groups because the overall prevalence of the outcome is not known. However, they can calculate an OR, since they can calculate the odds in the numerator and denominator with the information they have. ORs can also be calculated for cohort studies and RCTs, but RRs are typically preferred, in part because they are more interpretable.
As mentioned in the quote you provide, under some conditions RRs and ORs converge. Another reason ORs are often used is because log(OR)s are the natural output of a logistic regression. So if study performs a logistic regression analysis, it may simply report an OR because this is what they (almost) directly obtain.
For a more detailed treatment of this topic, this paper is a good resource.
