I think Wayne's explanation goes pretty far to resolving the OP's confusion, but I think there's one other clarification about SQL Server's internal workings that might help.

SQL Server creates and stores data in "pages" of about 8K each. Rows of data from tables, indexes, etc. are stored in these pages. SQL Server reads pages from disk into the data cache (RAM) (or creates them there) when required by the T-SQL statements being executed. SQL Server performs operations on the rows in those pages in the data cache and writes information about those operations to the transaction log. As a result, a page in the data cache may differ from the version of that same page on disk - a so-called "dirty" page. Every so often, "dirty" pages are "flushed" to disk, i.e., the changed pages in the data cache are written to disk.

For both a temporary table and a table variable, SQL Server creates the object in tempdb, but this only requires the creation and allocation of the necessary pages in the data cache. As long as SQL Server has enough RAM available, SQL Server can hold the pages of the temp table/table variable in the data cache and does not have to (but apparently may) write them to disk. If the amount of data stored in the temp table/table variable exceeds the amount of RAM available to the data cache, the temp table/table variable "spills over" and SQL Server must write those pages to disk, where they will live until SQL Server reads them into the data cache again. Of course, local temporary tables and table variables are dropped when the creating process terminates, whether the pages exist on disk or in the data cache.

So temporary tables and table variables always "use" tempdb in the sense that they are created there, but their data pages may exist only in the data cache. While Gail Shaw's article cited above demonstrates that temp table/table variable pages MAY be written to disk in the absence of memory pressure (the point she was trying to prove in rebutting the "table variables are memory-only" myth), it doesn't prove that they MUST be written to disk (which is not what I think Gail was trying to prove). Without knowledge of the underlying code, though, I can't say when or how SQL Server decides to write temp table/table variable pages to disk when there is no memory pressure.