No, the weight of your neuron does not factor into how your neuron votes. For instance, if your neuron follows one neuron with 1,000 ICP staked that votes "yes" and two other neurons with 50 and 20 ICP staked that both vote "no," your neuron will also vote "no." This is because neurons can be made to follow each other in various ways such that they vote automatically, representing a form of liquid democracy.
It's important to note that the voting power of your neuron, which is determined by the number of ICP utility tokens that are locked in the neuron, the neuron’s dissolve delay, and the neuron’s age, does influence the strength of your neuron's vote and how much reward you get per vote.