1. Teaching people who specialise in context areas (chemists, engineers) a pragmatic way to make a decision from data in the presence of randomness.
2. Automating analyses. (remember your Benjamini-Hochberg)
I'd never actually use it myself. Jeepers, just look at the picture.
also, p-values are used to determine the number of effects in a mixed model, or the number of factors in an ANOVA, etc. Sometimes confidence intervals on abstract parameters can be a bit less useful.
http://www.bmj.com/content/348/bmj.g2130
This is a secondary issue. For most non-stats people, I wouldn't recommend the confidence interval as a measure of clinical significance. I'd rather look at the actual distribution of the two effects and view quantiles of the predictive interval.
And, yes, in hypthoesis testing, p-values and confidence intervals are generally interchangeable. All they speak of is the overlap of the sample distribution of means, really.