"DC brushless"??? How can that be? Well, not your ordinary DC - more of a hybrid DC / 3 phase. In short: a PM rotor and multi-phase stator. An armature position sensor signals a controller to rotate the current in the stator windings. More here:
I didn't look at their AC motors, but AC brushless is an induction motor. Single phase cannot use an inverter (fixed speed). An inverter can be used on 3 phase, but is not required. *Some* form of 3 phase power is required. DC brushless always requires a controller, but not your typical VFD.
A brushless DC motor is essentially an inverted DC motor. Normal DC motors have a field on the outside (either coils for big motors or permanent magnets for PM motors) and a winding in the center rotating bit, the armature. The brushes conduct power to the armature via a commutator, a banded or slotted copper tube connected to the armature. As the motor spins, the commutator essentially reverses the poles on the fly, allowing the armature to alternatl\ely push or pull at the fixed field, creating rotation.
In a brushless DC motor, there is no commutator. The armature is the fixed magnetic structure and the windings are on the outside. Something needs to tell the controller the position of the armature so it can adjust or regulate the windings. This is typically a hall-effect sensor, so most brushless DC drives require that feedback, although more sophisticated new models can sense position without feedback. These wires are usually called Hall A, Hall B and Hall C, with one or more common leads. Thus, a brushless DC drive is a bit more complicated than a regular DC drive. The advantages to brushless are: no wearing brushes, elimination of drag from the brushes, and ability to reasonably sense position and speed, although not with the accuracy of an encoder for feedback.
In a brushless AC design, the speed is proportional to frequency, unlike DC drives where is is proportional to voltage. You would need a three-phase system and a VFD to take advantage of these motors for speed control. Some inverters do allow single-phase inputs and can control three-phase motors - look around and you'll find them, although they are much less common and more expensive due to the additional electronics involved.
I guess that depends on how you're driving them and what power sources are available. Both of these systems can be driven by servo drives or more basic variable-frequency drives (AC) or pulse-width modulating drives (DC). The sophistication of the drive means more to you than the motor.
For precise position control go for AC servo, especially if you have three-phase power available. Be warned that these are expensive and require a bit of knowledge (and software) to set up. Look on EBay and you'll find many makes and sizes of both drives and motors. They need to match in many ways: power output, feedback device, cabling, etc.
For speed control only you can use either an AC motor with a VFD or a brushless DC and a PWM drive. For overall speed control and power-to-weight ratio I'd go with the DC.