A 1 GHz Socket A Athlon IS a TBird. "Thunderbird" was the code-name for the Socket A Athlon before it was released. AMD simply calls them "Athlons" now, while the enthusiast market calls them "TBirds" to differentiate them from their older, slower Slot A counterparts.
Like has already been stated, the biggest difference besides the package (the majority of TBirds are Socket A vs. all classic Athlons in Slot A) is the cache. TBirds have full speed on-die L2 cache, whereas the older Athlon classics had off-die cache that operated at a fraction of the CPU speed (1/2, 2/5, 1/3).
If you have a Socket A motherboard, a classic Athlon is not an option for you. Likewise if you have a Slot A, a socket A TBird is not an option. There are some Slot A TBirds, but they're not that common.
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