This answer intends merely to expand slightly on the nice ones already given and summarize some of their content, in order to give a less expert reader an idea of what will be found in some of the references before consulting them.  My apologies if memory has faded the details following into serious errors.

The original question as asked, is answered already by Wirtinger, e.g. in his classic work Untersuchungen uber Thetafunctionen, on page 113, section 54, Uber die Thetafunctionen von zwei Variablen, lines -13,-14.  He says (possibly very roughly):

 “Each system of “eigentliche” (virtual?) everywhere tangent conics to a plane quartic curve C defines a Kummer surface, on which the curve C lies.  There are thus 63 such Kummer surfaces.”

The point is that a plane section of a Kummer surface determines not just a genus three curve but a connected double cover of it, for which the Kummer surface is associated to the corresponding Prym variety, and a genus 3 curve has 2^(2g) -1 = 63 non trivial double covers.

In general, there are (rough) correspondences between any two of the following data:

1) a curve of genus 3 equipped with a double cover;

2) a curve of genus 3 equipped with a half period;

3) a plane section of a Kummer surface;

4) a principally polarized abelian surface containing a genus 5 curve with twice the “minimal” class, i.e. a divisor in the system |2.(Theta)|.

5) a (semi stable, even) P^1 bundle over a genus 2 curve.

E.g. a (not necessarily effective?) everywhere tangent conic to a plane quartic determines a half period; a half period determines a double cover of a section of the trivial line bundle on the curve; a plane section of a Kummer surface inherits a double cover from that of the Kummer; a double cover of a genus 3 curve induces a P^1 bundle over the genus 2 theta divisor of the associated Prym variety via the Abel - Prym map; a P^1 bundle on a genus 2 curve yields a symmetric curve in its Jacobian which parametrizes effective twists of an associated rank 2 vector bundle.

These statements from memory will surely require correction by current experts, but may be useful as a sketch.

In addition to the references above to Wirtinger, Verra, and Birkenhake-Lange, one may consult Narasimhan - Ramanan: Moduli of vector bundles on a compact Riemann surface, Annals of Math (1969), and perhaps:

http://msp.org/pjm/1999/188-2/pjm-v188-n2-p09-s.pdf

The reference above to D’Ameida, Gruson, and Perrin, appears to be for the case of double covers of curves of genus 5.