It helps if you are interested in competitive fellowships. However, there are some fellowships where there are more spots than applicants.
For academic medicine, it helps but not necessary (unless you want to practice academic medicine at elite institutions). However, keep in mind, prestigious academic institutions do not pay well compare to their private practice equivalents. According to Medscape's 2012 survey, for Internal Medicine, average academic internists make $128,000, while an internist who work for a hospital makes $163,000 and for a multispecialty group practice, makes $194,000. For Anesthesiology, academic anesthesiologist make $220,000 while a private anesthesiologist in a multispecialty group practice makes $396,000. For Emergency Medicine, academia pays $159,000 while hospital-employed EM makes $247,000. Orthopedics, academia pays $225,000 but a single-specialty group practice makes $391,000
(source is Medscape 2012 Compensation Report, with 24,216 respondents across 25 specialities)
The "connections" will be helpful if you are interested in academic medicine, but once in private practice, those "connections" won't be as useful. In fact, it might be harder finding a private practice jobs since your academic attendings might not have those connections with private practice groups (since their main focus is on academia and academic medicine/research/QAQI projects, grant writing, etc)
There is no "big law" equivalent in medicine. And insurance companies/medicare/medicaid will not pay you more or reimburse you more if you went to MGH or Mayo or Hopkins compare to your colleagues who went to rural community hospital. An appy will be reimburse for an appy, whether you were train at a community hospital, or at Hopkins. And training at an elite institution will not protect you from lawsuits
And right now, there is such a huge physician shortage (in PCP and also in specialties) that patients are waiting months just to be seen - so while a few patients will specifically seek out "the best" (I want Harvard-train, Mayo-educated, Hopkins professor) (hint: you want to avoid those patients if you value your sanity and time), most of your patients just want to see a good doctor.
As for recruiting, it helps but it won't overcome other factors (if you are weird, you don't get along with colleagues, poor bedside manners and rude/impolite to your referral sources, don't complete dictations on time, etc).
Up until now, the process has been competitive (getting into a competitive college, doing well, getting into a competitive med school, doing well, getting into a competitive residency at a competitive hospital, and surviving residency/fellowship, and ultimately passing boards) - there should be a big reward for jumping through all those difficult hoops and beating your competition. And yes, while having a tenure clinician-educator tract position at Harvard Med (or even a tenure research scientist spot) is very prestigious, in the end, it is the doctor with the most business sense, in a practice that is the most efficient at procedures, scheduling, payor-mix, that will make the most money.