Refresh and download mean the same thing in this instance as long as you have scripted the application correctly.
As for speed it would depend on how much RAM and what processors are loaded on the server.
In most circumstances a servers processing power and amount of RAM are far greater than a desktop PC
If you have a qvf load script that runs manually ok in Desktop, then with Enterprise you can set it to auto run to a schedule. With the Caveat though that the Enterprise server must have the same access to the online database as your database - often server access to the internet is more locked down on a server than a desktop.
Regarding your Desktop lagging with large datasets vs. how Enterprise behaves, the answer is the infamous "it all depends". If it lags because your Desktop is low on RAM and paging, and your Server has oodles of RAM then probably yes. If it lags because of poor data model design and inefficient expressions, then no.
Qlik Sense Enterprise does have the ability to automatically refresh data from any data source. That means it will run your data load on a predefined schedule (and users will be locked out of their dashboards for the duration).
In my company's case, Qlik Sense Enterprise is deployed on a cloud-based VM. Therefore, we can control and scale the resources allocated to that VM to ensure it is running fast for the number of users. Remember, the Qlik Sense Engine uses IN-MEMORY COMPUTING, so make sure your hardware has a decent amount of memory for large data sets. Qlik Sense Enterprise uses the same engine as Qlik Sense desktop so it is not inherently faster than Qlik Sense Desktop. What makes it faster is giving it more resources.