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The Second House of Men spoke an unrelated language, the Haladin tongue. This language and Taliska were not mutually intelligible.
The exact origin of Taliska is not clear, but certain is that there are both Elvish and Dwarvish (Khuzdul) influences, suggesting the Atanatári (Fathers of Men) had contacts with both peoples before arriving in Beleriand.
Other versions of the Legendarium
Taliska was an early language developed by Tolkien. Unlike his later languages who are 'original', it was based directly on the Germanic languages, and has a lot in common with the Gothic language; this shows that Tolkien tried to connect his mythos with the origins of Europe, something that was evident in the The Book of Lost Tales. Gothic was an early interest of Tolkien.
It is possible therefore that Taliska belongs to an older phase of his creation, put along with Qenya, Ilkorin and Gnomish. In an earlier version, the name "Taliska" referred to the language of the Houses of Bëor and Haleth. The House of Hador spoke another language, which would eventually be conceived as Adûnaic. In an early text by Tolkien, some words he labels as Taliska refer to terms used by the Haladin.
However later, in The Peoples of Middle-Earth "Of Dwarves and Men" Tolkien wrote that the Bëorians and Halethians did not speak related languages: The Hadorians spoke "ancient Adûnaic", and the Bëorians had a closely related language, while the language of Halethians was unrelated to them. In this later conception, we can't know what would happen to the name "Taliska", if it should still refer to the language of the Haladin or of the Hadorians-Beorians, who, in the later conception, are now unrelated.
The name, as in this article, is still used sometimes to refer to the common language of the houses of Beor and Hador (instead of Haleth), but this usage is unofficial and perhaps mistaken.
A rather complete grammar and syntax of Taliska is known to exist, but despite work by Tolkienists this has as of 2008 not yet been published.