My relatives just stayed at The Henry over the US Thanksgiving week-end last month and were quite impressed vs.
The Henry is the better hotel of the two, recently was remodeled and updated and has very attractive pricing. Both hotels are within walking distance of the mall (essentially, just walking across the mall parking lot). There are several hotels near the Fairlane Town Center the two most convenient ones are on the outer edge of the mall property (but not connected directly to the mall): the Hyatt Regency and The Henry (which is a former Ritz-Carlton and is part of Marriott's signature collection of hotels). From the North Terminal, take the #280 bus north to Michigan Avenue, then transfer to the #200 bus to the Fairlane Town Center (be sure to ask the driver where to exit the bus). Depending upon your arrival terminal at Detroit Metro Airport, it may also require additional wait time and a shuttle bus to the North Terminal, which is the originating point for buses. Travel time is at least one hour each way and requires a change of buses.
The Fairlane Town Center in Dearborn is the closest mall to the airport and it is reachable by public transit (bus). Re hotels, all of the malls have hotels nearby. A car would give you a lot more flexibility re where to shop, stay and eat. I strongly recommend that, if possible, you consider renting a car.