It depends. The QuickShuttle is usually quicker and less expensive than Amtrak, Amtrak is generally more scenic. The QuickShuttle is MUCH more likely to be on time, and leaves right out of SeaTac, whereas the Amtrak station is more convenient to the downtown hotels, and, quite frankly, you need to budget substantial extra time for delays. (Also, be aware, the train station is all torn up for renovations, and it is difficult to find the entrance.)
The QuickShuttle is more convenient to hotels near the Space Needle.
Amtrak wheelchair accessibility, ahem, stinks. (If this were not a family board, I would use another word

) If you have a wheelie in your party, don't even consider it.
Remember that with Amtrak, you will usually need to pay for transportation from the station to the hotel. The QuickShuttle will take you to YVR where you can pick up transportation to the airport hotels, or it will take you directly to most of the popular hotels in Vancouver--this is both an extra savings and an additional convenience. The QuickShuttle will also pick up and drop off directly from the port.
I believe you do not need to remove your luggage at the border with Amtrak, whereas you do with the QuickShuttle (unless you are in a wheelchair). You show your passport at the border, they scan your luggae, and they reboard the bus with you and your luggage, and continue on to Vancouver.
Both the QuickShuttle and Amtrak have a "fast track" at the border, so long delays will not slow them down as much as a car.
The QuickShuttle has no food or drinks available for purchase, and it is a long ride. If you take it from the airport, I would recommend getting a take-out meal and some bottled water, or at least some snacks and bottled water. Amtrak (as I recall) has food available. Get something from the port to take on the QuickShuttle to go home.
If you are desperate, and wind up at Sea-Tac before 8:30 a.m. on your cruise morning, take the QuickShuttle. They will drop you off directly at the cruise terminal, and you have a good chance of making your boat. (I don't recommend this as a plan--too much can go wrong;this is only if you're STUCK doing this.) Don't ever take Amtrak if you're pressed for time.
One quick hint--if you're taking the QuickShuttle from downtown, take a taxi to the Best Western Executive Inn, and pick the QuickShuttle up there. It's less expensive to go from the Best Western than Sea-Tac, and it's a much shorter taxi ride from downtown.