Piquer vs. mordre
In French, distinguishing between piquer and mordre when talking about insects and arachnids depends entirely on the anatomy of the animal and how it attacks. Generally, piquer is used for a piercing action like a sting or a needle-like puncture, while mordre is used when a creature uses jaws or mandibles to clamp down.
Piquer
The word piquer means to sting or to pierce the skin. It is used for insects that attack with a stinger, like bees or wasps, as well as for bugs that pierce the skin with a needle-like proboscis, such as mosquitoes, bedbugs, or fleas.
Un moustique va me piquer si je reste dehors ce soir.
(A mosquito will bite me if I stay outside tonight.)
La guêpe a piqué mon chien directement sur le nez.
(The wasp stung my dog directly on the nose.)
J'ai peur que cette abeille puisse te piquer.
(I am afraid that this bee might sting you.)
Les puces ont tendance à piquer les jambes et les chevilles.
(Fleas tend to bite the legs and ankles.)
Le scorpion est connu pour piquer avec sa queue venimeuse.
(The scorpion is known to sting with its venomous tail.)
Il faut faire attention car les tiques peuvent piquer lors d'une promenade en forêt.
(You must be careful because ticks can bite during a walk in the forest.)
Mordre
The word mordre means to bite, and it implies the physical use of jaws, teeth, or mandibles to grab, pinch, or chew the skin. In the context of bugs, it is used for creatures like spiders, ants, or centipedes that chomp down rather than piercing with a stinger.
Cette grosse araignée a essayé de me mordre quand je l'ai touchée.
(That big spider tried to bite me when I touched it.)
Les fourmis rouges peuvent mordre très fort pour défendre leur colonie.
(Red ants can bite very hard to defend their colony.)
Si tu déranges un mille-pattes, il risque de mordre pour se défendre.
(If you disturb a centipede, it might bite to defend itself.)
Certaines chenilles utilisent leurs puissantes mandibules pour mordre.
(Some caterpillars use their powerful mandibles to bite.)
J'ai vu la mygale mordre le bâton en bois que le guide lui tendait.
(I saw the tarantula bite the wooden stick that the guide offered it.)
Les termites ne vont pas te mordre mais elles détruisent le bois de la maison.
(Termites will not bite you but they destroy the wood of the house.)
Summary
To summarize, you should use piquer for insects equipped with a stinger or a piercing mouthpiece, such as bees, wasps, and mosquitoes. Conversely, use mordre for bugs and arachnids that possess jaws or mandibles used to pinch or chew, such as ants and spiders.