If Tables Public and Tableau Desktop are installed on the same workstation, they have no interaction. Tableau Public will remain free and functional; Tableau desktop will require a licence key at some point, and the trial version will expire.
Tableau Public does not require any additional software. It may be used to publish to Tableau Online's public servers. However, you do not need to install anything locally, and you do not have much control over the public servers.
Many people share intriguing Tableau visualisations on public.tabeleau.com, which is essentially a massive archive of free Tableau visualisations.
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