Why do some aircraft have symmetrical tip airfoils while others do not?
Camber shifts the airfoil's section of lowest drag of the drag polar to higher lift coefficients. In order to create the least amount of drag in level flight, an airplane wing benefits from moderate camber. Most of the lift is created in the mid section, but enough lift is left towards the tips to justify camber there, too. In case of elliptical or tapered wings, the design lift coefficient for lowest drag at a given lift will be equal over the whole span (elliptic wing) or even peak near the tip (highly tapered and untwisted wing with no regard given to stall behavior or spar mass).
Normally the tips carry less load than the inner wing, details depending on wing taper and twist. This helps to prevent the tip section stalling first at high angle of attack. Good aileron effectiveness also requires an airfoil which equally produces positive and negative lift at low drag.
One example would be the Me-262. When Ludwig Bölkow designed the wingtip, very little was known about transsonic effects and a thin, symmetric airfoil with an elliptic nose was chosen because it showed a late onset of transsonic effects in wind tunnel tests.
When North American later designed the F-86, the outer wing design of the 262 was copied over because it had worked well in the German design.
Many Horten flying wings used symmetric tip airfoils because the bell-shaped lift distribution needed for stability resulted in a slightly negative lift at the tips at low angle of attack and very little positive lift at high angle of attack.
With some designs I am sceptical that a symmetric airfoil has been used even if it is claimed so in David Lednicer's outstanding collection. The Lockheed P-3 is there listed with NACA 0014 at the root and NACA 0012 at the tip - no, the Lockheed engineers 60 years ago knew better than that!