Ignoring dynamic links for the moment, the trust structure naturally supported by ERLs has an interesting and, from any publisher's point of view, highly desirable property: that you cleanly distinguish material of which you approve and in which you expect your readers to place some reliance. This may seem trite but is a growing concern, as in the laws of many countries a defamation suit may be brought against anyone involved in the distribution of a contested statement and not merely the author.
In the UK it is normal for libel litigants to sue and attempt to enjoin the distributors of newspapers and magazines with which they have taken issue; recently the Nottinghamshire County Council issued lawyers' letters and injunctions against a number of people who had links on their home pages to leaked copies of a report on satanic child abuse that the council considered to be its copyright [Notts].
So putting a link on one's home page can be dangerous; the controller of the referenced page might introduce controversial material and one could be sued. The implications in medicine include, for example, that a hospital which carelessly referenced a drug company's information page could find its standing in a negligence case substantially altered; equally serious consequences could follow elsewhere.
So the general use of ERLs rather than URLs would often be prudent practice, as the failure of a followed link to authenticate will indicate that it has been changed since the author of the link last consulted it, and he can thus in no way be held liable for its contents.
Other applications will typically arise where a publisher owes some particular duty of care, and we suggest some examples below.