While showing summarized info extracts are faster.
If you need to show detailed info using live mode is better since you are connecting tables with index columns.
Luciano Vasconcelos said it very well:
Speed. In my experience with Tableau Desktop, an extract is generally much faster, especially with large data sets. An extract will do the "heavy lifting" of large, complex joins and save them in a way they can be accessed in a flash. This is useful if you've made ODBC connections that require a password (you won't have to enter it except when refreshing). If your data is relatively small and simple, a live connection may be just as fast, but live connections to big data can be quite slow.
Speaking of passwords... Utility. Suppose your users don't have permissions to the ODBC sources you do. Using an extract will still allow them to view the snapshot of data without needing passwords.
Static Nature. An extract is a "snapshot" of the data, rather than a live view. You need to refresh the extract to get the newest snapshot. With a live connection, each time you open the viz you should see the latest data.
I'll throw in one more consideration:
Size. Because using an extract creates a second data type (a .tde), a single viz-- when paired with its .tde companion-- will take up more space. I find this a small trade-off for the processing efficiency I get in creating an extract.
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Another thing worth mentioning is that you'll need to extract any non-file data sources if you want them to be cached (as opposed to live) on Tableau Server after uploading them from Desktop.