They are both ways to get data to the cloud.
AWS is Amazon web Services which is the cloud services that Amazon offers. AWS give companies and individuals a number of ways to keep their data and programs and servers no longer a\t their site but on big computer farms that Amazon has. Now you can run your programs from anywhere in the world and your data is available from anywhere and you do not need all the servers at your place to handle your programs and data.
Some organizations have chosen to go 100% on the cloud, others a hybrid approach of some stuff on the cloud and some stuff still kept local and lastly some have chosen to ignore the cloud.
When an organization chooses to move all their stuff to the cloud to AWS that may be a major undertaking to get it there. They may have a large amount of data measured in terabytes (trillions of characters of data to put in simpler terms, think size of USA government debt) or exabytes of data (thousands of terabytes or possibly now in petabytes (thousands of exabytes). If it is just terabytes then it can probably be transferred via the network (Internet) to AWS. However as more data needs transferred the costs of using the bandwidth, the time it takes and having enough bandwidth to get there in a reasonable time.
This is where AWS Snowball and AWS Snowmobile comes in.
With AWS Snowball, AWS ships you a large stogie unit that you transfer all your data to and then ship back to AWS and they upload to your cloud storage. Basically you froze your data for a short amount of time so the name snowball. Quicker to transfer on each end on the sites.
However, if you have large exabytes or petabytes of data that exceeds the size of these devices. In this case AWS expands bigger and sends a tractor trailer if the large storage device filling up the trailer and you transfer data at your site to the storage in the trailer and then they get the trailer and take to their site and upload. This trailer and its equipment is Snowmobile which is a mobile device that you freeze your data in so this is a reasonable name. This approach is not cheap, but savings on time and all the bandwidth you would need and the costs associated with both.
One last AWS name that is similar is Amazon Glacier and that is part of AWS also but is where you save backups of your data, just as real glaciers hold ice and other stuff from long ago.