From an insert vs. hood point of view, typical manufactured inserts are designed to standard sized cabinets, and hence where a 36-inch hood might be used, the insert is only 34 inches or so. They tend to not be particularly deep, front-to-back. Otherwise, they can be functionally identical. Potentially, either could have in-hood, in-line, or external blowers. Recirculating, of course, limits one to in-hood blowers, unless a duct system is used to feed an attic or whatever filter bank before releasing the filtered cooking effluent back into the residence. I'm not aware of any such residential units; there are some large commercial recirculating hood systems.

From a recirculating vs. exhausting point of view, recirculating is limited by the quality and extent of the filters used to remove grease and odor. These will be partially effective, and the person here who has commented the most on the VaH ARS has noted that they are reasonable in grease collection performance. Perfect odor capture will likely be out of reach without the owner taking on the task of design and fabrication of an "ultimate" odor filter.

Recirculation has the advantage of not requiring a make-up air scheme or system.