Don't both use dimensions and fact tables?
Nope. Multidimensional uses Attribute Hierarchies and Measure Groups. Tabular uses Tables, and has no built-in notion of what a "fact" or "dimension" is.
Can't both have a star or snowflake schema?
Yes. And Tabular can have other designs as well. Tabular models can have single-table, or more normalized sachems, although using a star or snowflake design is generally considered a best-practice.
Don't both have measures and calculated columns?
MD does not have calculated columns. See Comparing tabular and multidimensional solutions
Also, if I'm using Power BI and I connect to SQL Server instead of SSAS, I can still do my thing with it.
Nope. Power BI always uses a Tabular or Multi-Dimensional model. When you connect to SQL Server with Power BI you are creating a Tabular model, and either Importing the data into memory, or creating a DirectQuery model (or a hybrid). In either case there is still a Tabular Model created, either embedded in the .PBIX or in a SSAS/AAS server.