While in bed, Peter sees the tarantula on top of his pajama tops. He says "Get it off me!" showing a close up of his face on his pillow. The shot goes to a close up of the tarantula and then back to Peter's head, but his hair is slightly different (slightly longer/bushier) and the pillow does not have the same creases on it. Also the lighting of the shot is slightly darker.
The spider changes type from scene to scene. Sometimes it'e not a tarantula.
The idea that a whole wing of a University would be renamed after the Professor after just 24 hours is absurd.