Snowflake and Big Query both are cloud data warehouse and although the architecture under the hood is little different, they solve the same purpose.
Then I cannot wrap my head around for why Snowflake is called as Saas and BigQuery is term by google as Paas.
As Snowflake was built purposely for cloud providers such as Google Cloud Platform, Amazon Web Services, and Azure, it requires almost no management or operational overhead same as how SaaS is being defined. It is a cloud data warehouse which can be hosted by the cloud platforms mentioned and passed as a distibuted software. Snowflake can also serve as PaaS, but PaaS requires more management (than SaaS) of the underlying hardware or software which Snowflake does not fall under.
BigQuery is categorized as PaaS since it is a native Google Cloud Platform data warehouse and a cloud-based platform service. It performs its function only on Google Cloud Platform and cannot run on any cloud provider, unlike Snowflake. PaaS delivers and manages all the hardware and software resources needed for application development which is being done when we are using BigQuery.
Basically, they just differ based on the scope of management and how each data warehouse is being managed. You can go through this documentation to help you better associate Snowflake as SaaS and BigQuery as PaaS.