In my opinion, customer service is king (specifically patient follow-up and office support). Technology is just the icing on the cake... but here are some distinctions that I've noticed with the technologies.
ANS - Represent the overwhelming majority of implants in my area. Constant Current system is the #1 feature. Patients who've tried both Medtronic's constant voltage and ANS' constant current feel a deeper and and more complete stimulation with the Constant Current system. #2 feature is frequency. ANS can go up to 1200 Hz. Medtronic can go up to 130 Hz. Read the literature and you'll notice that frequencies >250 Hz have been shown to recapture lost paraesthesia in CRPS patients and are often preferred right out of the gate in a trial. Big difference when you have a system that can't even reach a high frequency. #3 feature is simplicity. All of the ANS leads are compatible with both the conventional and rechargeable generator. A lot of Medtronic leads require an extension, adding an extra point of system failure.
Medtronic - Very few practices in my area still use Medtronic, and only do so because they've had a long relationship with the rep, which is understandable in my opinion. The only advantage to their system is they offer the intrathecal pumps. Whether's there's really a need for intrathecal pumps in the first place is highly debatable in my opinion.
ABi - Never made much of an impact in my area because the reps were rather inexperienced. Personally, I love the smaller generator (even with the shorter battery life), but their leads are inferior. ABi leads only cover 1 vertebral segment. ANS and Medtronic cover 2 segments. When you consider lead migration and neuroplasticity, that's too much of a liability in my opinion. Otherwise, they do offer some of the same advantages of ANS (constant current, high frequency).