“To weight” is to assign a weight (e.g., to weight variables differently in a model), whereas “to weigh” is to observe and/or record a weight (as a scale does).

A few other cases of this sort of thing:

affect (n). an emotion or feeling. "She has a positive affect."

effect (n). a result or change due to some event. "The effect of her affect is to make people like her."

affect (v). to change or modify [X], have an effect upon [X]. "The weather affects my affect."

effect (v). to bring about [X] or cause [X] to happen. "Our protests are designed to effect change."

Also:

cost (v). to require a payment or loss of [X]. "That apple will cost $5." Past tense cost: "That apple cost $5."

cost (v). to estimate the price of [X]. "The accounting department will cost the construction project at $5 million." Past tense costed. "The accounting department costed the construction project at $5 million."