Well they're characterized as one of the 3 main summoning contracts in terms of relevance to the narutoverse. Like any summoning they are animals with a higher level level sentience and control over their chakra and are shinobi in their own rights as animals. The specific relevance of snakes are their links to the characters primarily the Sannin the second gen Sannin (Naru, Sasu, Saku) and Kabuto.
The Sannin, originally taken from the legend of "Jiraiya the Gallant" Japanese folklore with Jiraiya, Tsunade, and Orochimaru in the Narutoverse being the namesakes of the three main characters in the legend; Jiraiya and Orochimaru being shinobi having the powers of Toad and Snake magic (respectively) and Tsunade being a "princess" who could use slug magic. This again alludes to Tsunade's title of "Princess" given to her by Jiraiya, Danzo and the likes. The Sannin and the second generation of Sannin all use the same Toad, Snake, and Slug "magic" or in our case summoning techniques and ninjutsu as their predecessors.
Canon wise, they are the second animal linked to natural energy manipulation demonstrated with toads being the first. This meaning that snakes too possess the ability of Senjutsu and sage chakra/sage mode. This is realized through Kabuto's explanation in the series which also reveals the truth of Orochimaru's achievements in it, though noticeably inferior to the level in which Kabuto is able to attain it due to his infusion of Jugo's cells but it sheds light on the Curse Mark's of Earth and Heaven (and the variations used on the Sound Four and by Orochimaru in general). The curse marks being the sealing of chakra from Orochimaru's serpentine version of natural energy. Nevertheless natural energy, being nothing more the chakra of nature and those who enter sage mode are merely ones who gather this chakra into their body and harmonize it with their own, the reasons for why the various different versions occur (Snake vs. Toad or Hashirama's version, or Jugo's pseudo sage mode, etc) is still foreign to me. If the methods of balancing are indeed the same with the exception of Jugo clan members who can passively absorb, why do the characteristics differ between users like toad and snake? Does the contractual agreement between the signer and the animal have a relevant play in it even if the person's chakra is no different fundamentally than somebody who may use Toad Sage mode? I'm not sure, you can enlighten me there if you have some insight since you're the sensei.
Though back to the specific, the usage of snakes seems to be more versetile in combat as opposed to toads as snakes possess a more flexible form and usage. We've seen demonstrated snakes being used as basic bonding/restricting agents, easy substitutions, etc. However being used by contractors who show little concern for the animals themselves such as Sasuke and Orochimaru, both antagonist role characters seems to be a contributing factor as to why these animals are more used as tools rather than companions. As well, the lack of sentience from most snakes, with only two snakes in the series (to my knowledge) even uttering a word or appearing like a "comrade" in loose terms (Aoba - Sasuke's personal summon, and Manda - the now deceased boss) is also a contribution. The other two summons, Toads and Slugs have almost always had some sort of sync and team play with their summoners.
And that's it