There is a special case where it may be worth contacting the graduate department - more on that in a moment.
First, by being informed that you have been put on the waitlist for admission, they are saying that you have not been admitted at this time - but they aren't saying you are officially rejected yet. So you should indeed keep your options open to the extent that you can, but most of the time you cannot rely on admittance.
The reason they stated the date of April 15th is because in the US this is a special national date, where the vast majority of institutions have agreed (or been required by federal funding agencies) to allow anyone admitted with a funding offer to respond by April 15th without fear of losing their spot/funding by not responding right away. It's probably the nicest part of the application process for applicants, as if you receive multiple offers you can wait until you receive responses from all your preferred institutions before making a decision by the shared deadline.
The downside, of course, is what you experience with the waitlist - institutions are often in the position of not having heard back from all the offers they've made until very near the deadline, so they can't let waitlisted people know of their final status until past such a date.
Generally there is not said to be an advantage to contacting the programs to re-affirm your interest, especially at very popular/highly ranked/prestigious programs, because just saying "I'm still interested" doesn't change anything.
When You Should Consider Contacting Waitlisted Programs
The big exceptions to the general case is when something new has happened that isn't properly reflected in your application. The most common such case is when you have received an offer of a fellowship offering significant funding, such as the NSF graduate fellowship (or any non-trivial funding source of the kind). In such a case if you receive such a recognition, I'd encourage you to notify any programs where you are still in the waitlist. While there is no general rule about how programs handle such a situation, some seem to care and many people claim they were moved from the waitlist to acceptance after receiving such a major award.
However, given the extreme competitiveness of such major awards and the fact that they are often more competitive than PhD programs are themselves, this can't be relied on and there is absolutely no guarantee the program will decide to accept you with outside funding.
You could potentially also want to contact them if other major improvements have happened to your profile, such as receiving other national recognition, major important/prestigious publications, etc. This may be of limited usefulness as I don't know any departments that will reconvene an admission committee to discuss your new achievement, and I don't know that any department will just adhoc re-arrange the waitlist order based on such a thing. If you aren't super annoying about it I suppose it couldn't hurt. The potential here is further limited because anything that could result in such a situation should have already been talked about in your application materials (including in submission/preparation papers, the experience that led to such papers, etc).
Regardless, if you choose to contact the department I'd personally advise you not mention the "if it doesn't work out this time I'm giving up on the whole thing". I don't think mentioning that will help your situation any, even if it's how you feel. YMMV.
Above all, any contact you make should be respectful and professional. Repeated calling (hour after hour, day after day), email after email, or anything similarly negative could actually zero out your chances entirely. Reasonable timely correspondence will not carry any such danger.
