First they are completely different slides (side mount vs under mount). I've used both. Blum IMO is the standard for drawer slides. That being said, not all projects call for an under mount slide.
Benefits - Blum Tandem are smoother operating and less prone to drag from my experience. With side mount drawer slides, the margin for error on squareness and drawer width is less. For example if your carcass tapers toward the back 1/16 you could have a drag issue with the side mount slides. I have not had that problem with the Tandems.
-Better soft close mechanisms on Blum slides.
-You can't see the Blum Tandem slides when installed.
-3/16" vs 1/2" With side mount you lose more drawer space. You need a full inch (1/2" each side) to account for the slides. With Tandem the drawer width is 3/16 off each side.
-Side mount are easier to install. With Tandem it is handy to have the jigs for drilling out the back of the drawer and when installing the locking mechanisms. A screw hole slightly at the wrong angle can cause the locking mechanism to not sit flush with the bottom of the drawer. The back end of the slide has a hook that goes into a hole at the back of your drawer..if this hole is off, your slide doesn't fit on the drawer perfectly and can affect the adjustment, fit and operation. Using the jigs eliminates these mistakes (90% of the time it works every time!
-Tandems are more expensive and have more parts. They also require more drawer modification. Side mount do not require any extra parts typically, and require no drawer modification. These modifications need to be considered before you build the drawer. I believe the dado for bottom, for example, needs to be 1/2" up for the slide to fit correctly. Side mount...no such requirement.
I could go on, but I will stop boring you. Both slide manufacturers are excellent IMO. I use them both, but for most projects I use Blum Tandem.
Hope that helps,
Nick