Can anyone explain the benefits of using one option over another?
Once you get over the "tactical ugliness" of a picatinny rail and tactical rings, the
function will win you over.
You can easily gain more elevation travel and most do with a standard 20 MOA base (and have a much stronger setup than Burris rings with inserts) and they can be made in custom slopes for whatever you want.
No more worrying about tube length, action length, eye relief, etc, compatibility. You can put the scope very far forward or very far rearward--you can place it in the "correct" position for you.
You can swap scopes from rifle to rifle very easily. Many say they never do this, but often that's because doing it with standard rings is such a PITA. Once it becomes easy you may find yourself doing it a lot.
You can have a backup scope sighted in and ready to bolt on at any time that you can take with you and keep in camp.
You can put a torque wrench on the mounting screws to verify everything is tight for some peace of mind without losing zero. Most standard rings pretty much require disassembling everything to check the base screws and/or can't be counted on to return to zero very well. The end result is with a standard setup many people will let many YEARS go by without ever checking because once the scope is on the rifle they don't ever get up the motivation to take it off.
Strength. A picatinny rail (especially with #8 screws and/or glued to the receiver) and quality tactical rings are a huge jump up in strength from most typical standard rings. Long range guys typically like big, heavy scopes. Couple that with a hard kicking rifle and an "overkill-strong" mounting system is an ounce of prevention that can save you much hassle in the future.
Speaking of ounces, due to good design, some high quality rails and rings made from aluminum and/or titanium can still provide this massive increase in strength with only a very small weight penalty over old-style standard steel rings/bases.