Symbol("a") == Symbol("a") is more-less equivalent to Common Lisp's (eq (make-symbol "a") (make-symbol "a")), both of which will return false, because (in Lisp terms) we're creating two different symbols. In Common Lisp, the default behaviour of a reader if it sees a stand-alone symbol token, is to intern it - put it in "global"[0] storage, so that if the reader sees the same-looking symbol token again, it returns the same symbol instance it previously interned.
In JavaScript, per this article, the interning behaviour is available through Symbol.for(), i.e. Symbol.for("a") == Symbol.for("a") should return true, much like (eq 'a 'a) would return true in Lisp.
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[0] - it's actually per-package storage, but that's getting into the more advanced details of CL.
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EDIT replaced gensym with make-symbol in Common Lisp example, as the former is used primarily in writing macros, so it ensures the created symbol has an unique printed representation, which helps make macroexpansions more readable.
(make-symbol "FOO") -> #:FOO
(make-symbol "FOO") -> #:FOO ; NOT eq to the previous one
(gensym "FOO") -> #:FOO103
(gensym "FOO") -> #:FOO104 ; note the incrementing counter appended by gensym
Compare http://clhs.lisp.se/Body/f_mk_sym.htm and http://clhs.lisp.se/Body/f_gensym.htm.