In general, print servers are used to both distribute drivers to client computer and centrally process and manage the print jobs. In large environments it's useful to have homogeneous drivers (which will usually contain certain printing configurations that are desirable to control centrally) in addition to having a central location for managing and logging print jobs. For example, the most common setting I see companies want to "push out" to client PCs from the print server is to default to black and white printing, rather than color printing (to save money on the more expensive color ink).
So yes, in the general case, the client computer will connect to the print server, acquire the printer driver from it, and then connect to the server to actually print to that printer. It is possible, though much less common, to connect to a print server just to get the right driver, install the printer directly, with that driver, and then bypass the print server by printing directly to that printer. But note that this is dependent on how the printer is installed on the client. It's either installed "directly" as a stand-alone printer on the client, or installed as a shared printer from the print server, and this is what determines whether the client connects to the printer directly, or through the print server instead. This is where the distinction between a physical printer ("print device") and a logical printer matters - it is actually possible to have the same physical print device installed multiple times as different logical printers. For example, by installing the same print device once directly, and once via the shared printer on the print server.
Since you tagged your question with Server 2012 R2, this Technet doc on Server 2012 Printer Sharing Technologies will probably be of interest. Note the section titled: Enhanced Point and Print, which is a technology that allows clients to print to compatible printers through a Windows Server 2012+ print server without installing a specific driver for the printer on the client. Meaning, of course, that it's also possible to use a print server so that clients don't need to install drivers for specific printers, but it's still most common that a print server will but distribute drivers to client and process/manage client print jobs.