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.
Taliska, unlike Tolkien's later languages, was based directly on the Germanic languages, and has a lot in common with the Gothic language, which shows that Tolkien tried to connect his mythos with the origins of Europe. Gothic was an early interest of Tolkien.
However it is possible that Taliska belongs to an older phase of his creation, put along with Qenya, Ilkorin and Gnomish but the name is still used sometimes to refer to the common language of the two tribes.
A rather complete grammar and syntax of Taliska is known to exist, but despite work by Tolkienists this has as of 2005 not yet been published.