Original Venom movie designs replaced Spider-Man’s symbol with a “V.” The first full visage of the titular anti-hero of the 2018 Sony film starring Tom Hardy, dates back to the 1988 pages of Marvel Comics Amazing Spider-Man #300. However, the original debut of the actual black Spider-Man costume is in Marvel’s mini-series, Secret Wars, issue #8 in 1984. Upon later discovering the costume’s true nature as an alien symbiote intent on completely merging with him, Peter Parker, aka Spider-Man, would discard the suit, leaving it to find a new host, Eddie Brock. The two join together to become Venom but retaining some of its features from its time as Spider-Man’s costume, most notably the giant white spider emblazoned on his chest.
Due to the original deal between Sony Pictures and Marvel, the Spider-Man character Sony holds the rights to was allowed to be featured in the MCU. However, that deal did not include many of the “Spiderverse” properties Sony owned, such as Venom. Thus Sony’s Venom film stood alone without mention of Spider-Man, with Venom’s origin being retconned in the movie. Venom, minus the web-head, went on to become a hit regardless, the success of which encouraging Sony to explore its own “Venomverse." Utilizing other members of Spidey’s rogues' gallery, which they had exclusive rights to (like Carnage and Morbius), Sony could build out their own MCU-like shared cinematic universe.
As uncovered by TheVenomSite.com, original Venom designs had the film potentially replacing the character’s spider symbol with a giant “V” shaped marking. Paolo Giandoso, who has worked on other films including Captain America: Civil War, Ant-Man, and Wonder Woman is the artist behind the visual concept alterations. Having removed Spider-Man as a critical and integral element to the Venom backstory, then, by default, removed the carried-over logo, typically featured prominently on the Venom suit. It's possible they felt the "V" was too distracting a reminder of what should be there, but for whatever the reasoning, they ultimately chose to go with no logo at all.
Cooperation between the film properties of Sony and Marvel has evolved since the original Venom film, as we’ve seen Spider-Man MCU references in Sony’s newest upcoming comic-adapted film, Morbius. The Easter eggs from the trailer now suggest that Spider-Man does share the same universe Sony is building; thus, the “Venomverse” and the MCU can be the same. There are even many rumors and hopeful speculation that the web-slinger may swing through the Venom sequel, Venom: Let There Be Carnage.
Venom has grown into an icon, independent of Spider-Man. Despite the logic behind the suit changes, the spider remains as much a defining aspect of Venom’s appearance as his razor-sharp teeth and serpentine tongue. Their "spider" connection only further embiggens the two of them, with Venom acting as a dark reflection of Spider-Man. Meanwhile, Spidey serves as an example of the hero that Venom could one day become, were he to let go of his rage and personal grudges. Hopefully, that indispensable spider symbol will one day be featured again on both heroes, inexorably binding them together, as any good webbing would.