In the movie Venom, instead of using webs, Venom used parts of himself to swing him to safety. And he also clawed his way up a tall building instead of using webs and "sticky hands".
The symbiote can mimic Spider-Man's powers, but does not necessarily duplicate them exactly.
Instead of given its host actual organic webbing (or, in the comics and the newer movies, mechanical webshooters), the symbiote uses its own body to mimic webbing.
It may allow the host to use feet and fingertips to crawl up a wall, as Peter Parker can do. However, since Venom's appearance includes claws, the host may simply prefer to use the claws to scale a building, instead of fingertips. Using the claws does seem in keeping with Venom's personality - it's scarier (and possibly allows an outlet for his more violent side).
I suspect that, if Venom was actually explicitly trying to mimic Spider-Man, he could indeed use his fingertips and feet to scale a building.
In the Venom movie, Venom has never encountered Spider-Man (who as far as we can tell may not even exist), and, as such, has no access to his powers or normal modes of doing things. The only reason Venom shot webs or wall-crawled in the same way as Spider-Man was because he was making himself useful by producing what he needed (webbing) and also, through the symbiosis, learned the tricks of Spidey's body and powers (wall-crawling, also managed to be immune to Spider-Sense).
It should be noted, that when Venom DOES shoot webs, he's doing pretty much the same thing as he does in the Venom movie - using part of himself to grab a distant object and swing to it. The only difference is, he's making it look like webs, which movie-Venom has no reason to do. It's possible movie-Venom could also learn the fine control to climb walls without claws, but there's no particular need for it if claws do the job, and claws are cooler - you don't look like Venom if "looking like a thirteen-year-old's view of cool" isn't part of what you're going for.