Tantum Ergo is the last two verses of Pange Lingua sung on their own, usually during exposition and Benediction of the Blessed Sacrament. Many times Pange Lingua is sung in its entirety on occasions like Holy Thursday, when the Blessed Sacrament is being removed to the altar of repose, or on Corpus Christi.
There are two Pange Lingua hymns. The one you cite, by St. Thomas of Aquinas, which was inspired by another, by Venantius Fortunatus. The latter is used as a hymn for the Office of Readings and Lauds of Holy Week, and is not related to the Eucharist but to the Cross and Passion.
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