The best way to see this independence is to read the original papers.
Turing's 1936 paper introducing Turing machines does not refer to any simpler type of (abstract) finite automaton.
McCulloch and Pitts' 1943 paper introducing "nerve-nets", the precursors of modern-day finite-state machines, proposed them as simplified models of neural activity, not computation per se.
For an interesting early perspective, see the 1953 survey by Claude Shannon, which has an entire section on Turing machines, but says nothing about finite automata as we would recognize them today (even though he cites Kleene's 1951 report).
Modern finite automata arguably start with a 1956 paper of Kleene, originally published as a RAND technical report in 1951, which defined regular expressions. Kleene was certainly aware of Turing's results, having published similar results himself (in the language of primitive recursive functions) at almost the same time. Nevertheless, Kleene's only reference to Turing is an explanation that Turing machines are not finite automata, because of their unbounded tapes. It's of course possible that Kleene's thinking was influenced by Turing's abstraction, but Kleene's definitions appear (to me) to be independent.
In the 1956 survey volume edited by Shannon and McCarthy, in which both Kleene's paper on regular experssions and Moore's paper on finite-state transducers were finally published, finite automata and Turing machines were discussed side by side, but almost completely independently. Moore also cites Turing, but only in a footnote stating that Turing machines aren't finite automata.
(A recent paper of Kline recounts the rather stormy history of this volume and the associated Dartmouth conference, sometimes called the "birthplace of AI".)
(An even earlier version of neural nets is found in Turing's work on "type B machines", as reprinted in the book "The essential Turing", from about 1937 I think. It seems likely that many people were playing with the idea at the time, as even today many CS undergrads think they have "invented" it at some point in their studies before discovering its history.)