The benefit of using multi select variants is that you can assign more than one value from your variant criteria to a single object.
Difference between single select and multi select variants
Multiple values: If an object can be used by several variations of your content you can assign multiple values to the variant - a value to match each variation of the content. You do this from multi select variant criteria only.
Single value: When an object can only be used by a specific variation of your content you should assign a single value to the variant. You do this from a multi or single select variant criteria.
While many topics in your content will need specific variant values assigned, there may be some topics or other types of objects that are applicable to more than one product, location, or other variant requirement.
Example: Using a single select and multi select variant
For example, Anne Ortha maintains user guides for the Acme Teleporters, with guides for each product (X1000, X2001, X3001, and X5000), and variations of the guides for most states.
Anne wants to create a topic that applies to the Acme Teleporter products X2001 and X3001 that will be used in the guides for New York, California, and Texas, but not for other states. This means the topic will only be used in the guides for X2001 for New York, X3001 for New York, X2001 for California, X3001 for California, X2001 for Texas, and X3001 for Texas.
Using single select variants, where only one criteria value for product and one criteria value for state could be applied, Anne would have to create a variant object for each combination - that means six objects. If the information changed, Anne would have to update each topic.
Now using multi select variants, Anne creates one topic and assigns all of the criteria to that object. This means that one topic will meet the filtering criteria for each of the six guides. If the content changes, Anne only needs to update one topic.
Filtering and publishing
When Anne filters for the object, publishes from Author or quick-editing mode, or uses the object in Review or Translations, she continues to choose the single criteria values for her job. This is because filtering uses single values so you'll assign the values for the variant you looking are for.
For example, Anne wants to publish the guide for the X3001 for users in New York. In Author, the filtering for the publishing job for this guide will use the Product = X3001 and State = New York (NY). The topic she created using the multi select variants will also be used in the job because, along with all of the other values assigned, the topic has the X3001 and New York values assigned.
State = New York, Product = X3001