There has been strong recent interest in testing interval null hypotheses for improved scientific inference. For example, Lakens et al. and Lakens and Harms use this approach to study if there is a prespecified meaningful treatment effect in gerontology and clinical trials, instead of a point null hypothesis of any effect. Two popular Bayesian approaches are available for interval null hypothesis testing. One is the standard Bayes factor and the other is the region of practical equivalence (ROPE) procedure championed by Kruschke and others over many years. This article connects key quantities in the two approaches, which in turn allow us to contrast two major differences between the approaches with substantial practical implications. The first is that the Bayes factor depends heavily on the prior specification while a modified ROPE procedure is very robust. The second difference is concerned with the statistical property when data are generated under a neutral parameter value on the common boundary of competing hypotheses. In this case, the Bayes factors can be severely biased whereas the modified ROPE approach gives a reasonable result. Finally, the connection leads to a simple and effective algorithm for computing Bayes factors using draws from posterior distributions generated by standard Bayesian programs such as BUGS, JAGS, and Stan.