Aspririn is far more convenient and safer to administer. Coumadin requires diet restrictions and periodic testing (PT/INR checked weekly or monthly) and occasional dose adjustments. If you accidently let your PT/INR rise above safe levels you could have serious bleeding problems and complications. If it drops too low, you are at risk of a stroke.
As Uboat said, there are risk factors (age, BP, prior stroke, etc.) that greatly increase your chances of clotting/stroke and so coumadin is indicated. Your cardiologist assesses these and determines if you can get by safely with just aspirin or if coumadin/warfarin need be prescribed. Here is how the medhelp cardiologist answered a similar question to yours.
http://es.
medhelp.org/posts/show/253811
BTW I've been on coumadin for over 5 years and would much rather be on aspirin. On the other hand, I don't want to get a stroke either and it's not too inconvenient.