I have both. A-Bolt II Medallion in .300 WSM and X-Bolt Medallion in .308 Winchester.
The A-Bolt is more of a traditional bolt action rifle. Removable spring loaded magazine by hinging the floor plate down. It is a bit tricky to get the magazine released from the floorplate, especially with gloves on but with practice it becomes easier. Bolt is locked down with the tang safety engaged.
The X-Bolt is Browning's answer to the Winchester Model 70 3-position safety lever. With the tang safety engaged, the bolt can be opened by depressing a release button near the bolt handle. This allows the bolt to be opened with the safety engaged. I like the feature, very easy and intuitive to use. It has a plastic (which at first seems cheap and cheesy) rotary magazine. This does have advantages, though, since it doesn't clank around in your pocket or vest and it will not corrode. It is also very smooth to load and for the bolt to strip a round from during loading. Magazine installation and removal is a snap.
Both rifles have a feature that is not noted often but I very much like as far as smoothness of the action. When the bolt is retracted, there is a spring loaded arm that keeps the next cartridge in the magazine from dragging on the bolt as it is being retracted. Once the bolt reaches the end of travel to the rear, the arm allows the cartridge to move up into position and be pushed into the chamber by the bolt. Extremely smooth and positive. This has to be felt to be appreciated.
I agree with other's sentiments regarding the styling of the X-Bolt preferring the styling of the A-Bolt. But the design, functionality, accuracy, ease of operation of the X-Bolt makes the styling moot in my opinion. They are both outstanding rifles and I would not hesitate to consider either model if I happened to find one in the caliber and condition of choice.
Dan