The benefits of the Left-side, non-reciprocating handle should be fairly obvious (at least for righties). You can maintain positioning on the rifle, including keeping your sight picture/view on target and manually operate the rifle as needed without altering your position much at all unlike a traditional CH which generally requires you to contort yourself at least somewhat around the rifle. Your dominant hand maintains control over the rifle the entire time, including positioning on your grip/controls for the rifle/trigger/etc. Also, with respect to the JP design, the folding lever/latch of the handle is nice because it gets the handle out of the way where it isn't as prone to snagging on everything coming and going. The handle on the ASA also folds and is non-reciprocating, but it is not quite as flat/streamlined as the JP.
As to the right-side, reciprocating designs like the Fulton Titan, etc., they are, again for righties, strong-side only which requires you to come off the rifle with your grip hand/trigger finger thus giving up some positioning on the rifle. Furthermore, because the CH is a fixed prong sticking out of the side of the rifle's carrier that doesn't fold or attempt to get out of the way, it can get hung up on stuff, jammed into your back/side/etc. when carrying, as well as moving back and forth during battery which could, in rare instances I can't generally see happening, pose problems for some folks.
Finally, and while I've tried to be nice and not say anything too negative re: ASA...they don't have the most stellar track record. The company has changed hands a number of times I believe and sometimes...for the worse. They seem to have their @#$% together in more recent past, but for a while at least, the quality, reliability, accuracy, etc. of their products was lacking and their support/CS was...well...non-existent. That left a bad taste for them for many folks and only time will tell if they are on the straight and narrow now. They certainly have some innovative designs/products coming to bear in 2014, but again...only time will tell.