Some info on the SUA You asked about.
Springs
Under: With the springs swooping under the axle they generally need more arch to achieve lift and can actually stiffen the ride and hinder the flex. Yet one of the great positives about this setup is that your center of gravity remains lower and often offers a more stable feel on off-camber obstacles.
Over: Since you are gaining lift by moving the perches on top of the axletube, a flatter or softer spring pack can be used to gain articulation and create a more cushioned ride. This also opens you up to more spring options, and for the budget-minded, this means taking a trip down to your local junkyard and trying out different spring combos.

Shackles
Under: A small shackle lift of 1 to 2 inches is an inexpensive way to get your Jeep a little height without stiffening the ride or hindering the flex, but don't overdo it. An overly long shackle can create clearance, handling, and stress issues that are not worth the lift.
Over: Depending on whether you keep the shackles in the front of the Jeep or perform a shackle reversal, longer shackles are not really necessary. On the front axle, swapping the shackles to the rear will help your rig's forward movement over obstacles, but will cause your driveshaft to extend farther during articulation. A basic spring-over conversion does not require you to reverse the shackles, though many wheelers opt to.

Axlewrap
Under: Without a doubt the strongest argument for keeping the springs under the axle is the limited axlewrap that's received compared to a spring-over. It's all about torque forces. While you may not have the ground clearance that a spring-over has, your axle will be held more firmly in place and able to deal with the loading forces more easily.
Over: Almost every SOA configuration has axlewrap. How much axlewrap depends on the softness and arch of the spring. Lift blocks will amplify wrap, and we must suggest that if you are planning a spring-over, pencil in some sort of wrap device. Don't worry if you are fab-challenged, as many aftermarket antiwrap bars are available.

Fabrication
Under: Basically little to no fabrication is needed, as most aftermarket kits are designed as bolt-on units.
Over: Depending on your skill level you can have a couple options. The first would be creating your own homebrewed spring-over kit. This is completely fine as long as you feel confident in your welding and fab skills. Another option would be ordering a do-it-yourself kit from one of the many aftermarket manufacturers. While most require some mechanical talent, they can be done in your driveway with basic tools and a welder. Some are even bolt-on.

Driveline
Under: Depending on lift height (usually limited to 4 1/2 inches) and transfer-case model, you can get away with running a transfer-case drop. In later models a new driveshaft and/or slip yoke eliminator may be required.
Over: Since the spring-over usually creates close to 5 inches of lift with stock springs, new driveshafts and/or a slip-yoke eliminator are generally needed.

Clearance
Under: Many of the SUAs have the U-bolts facing the ground, and since that means the spring plates are on the bottom of the axle, components are subject to increased
off-road damage. The upside is that quite a few aftermarket companies make U-bolt flip kits that can take care of the down-facing hoops.
Over: Getting the springs on top means flipping the U-bolts to face up. This gives your axle more clearance for the suspension components and other undercarriage vitals.

And There's More...
Here we've touched on just a few of the larger pieces that you'll need to consider when deciding for or against performing a spring-over conversion. There are always those basic elements that will need attention as well: brake lines, steering linkage, track bars, and sway bars, to name a few. If you are thinking about purchasing a kit, whether it be your basic spring-under lift or a fully weld-on SOA, calling the manufacturer and asking questions to make sure it comes with everything you need is always a sound decision.