Peter Parker is absolutely more spider-like than they go into.
He most definitely has fangs that are sharper than the rest of his teeth. They're hollow and contain venom that paralyses people for hours. It's incredibly strong.
His senses increased exponentially. He can hear a criminal from ten blocks away. There's a fly that won't leave him alone? That thing is smacked with the precision of Hawkeye. There's chemoreceptors on his hands and feet that allow him to feel the microscopic dents on anything. The only sense that seems to have remained the same is his sense of taste. He has olfactory receptors, too.
There are spinnerets on his wrists that are hidden under skin until opened on command or massaged out. Yes, he has organic webs. They are incredibly strong, but Peter developed an acid that can dissolve them over a thirty-minute timeframe that his suit expels using the shooters.
One day, a robber catches Peter off guard and hits him with a machete. Hard. Peter finishes up the fight, and doesn't realize anything is wrong until Tony shows up, completely panicked. Peter looks down at his shoulder, and it's hanging on by a small chunk of flesh still attached to him. His skin is visibly knitting itself back together. Tony is gobsmacked, and Peter just says, "Oh, that just happens sometimes." and moves on. The arm reattaches itself before Tony can force him into the Med Bay.
There is no place that Peter cannot reach. He will climb on walls for fun. Sometimes, he even go from floor to ceiling in a second when he's scared, on instinct.
Spiders love high places--for multiple reasons. There's lots of anchors for webs, rising warm air, and a steady supply of insects for food. Peter loves high places because they're safe. In the corner of his room in Avenger's Tower, there's a knot of webs that he sleeps in after particularly traumatizing events or tiresome patrols.
And of course, sometimes when he webs up a bad guy, and the bad guy is just wriggling in the web trying to escape, Peter has the instinct to eat them. But only for a minute!! It's totally normal and he wouldn't actually eat them!! That's weird!! Totally....
