Dracula is a Gothic novel strongly influenced by the Enlightenment aesthetic of the Sublime. This is particularly noticable through Stoker's depiction of the weather, in itself a common theme in the Sublime.
Stoker typically shows the weather like a symbolic character, using it to produce atmosphere and emotions in an almost cinematic way. Both good and bad weather are used in this way. The weather can also provide a neutrality, particularly in depictions of the night. As in most horror tropes, the night is implicitly a time of danger, so unless there is need of a additional emphasis Stoker doesn't use his extra character.
Dracula opens in Budapest with Jonathan enjoying pleasant weather` a literary symbol of calm before the storm. However, as Jonathan progresses towards the Castle, the weather becomes increasingly intense, helping to increase the tension in the reader.By the time Jonathan arrives at the path, the weather has become a dramatic thunderstorm.
In Whitby this sequence of calm then storm is repeated. The neutrally pleasant weather at the beginning allows Lucy and Mina to enjoy walks along the clifftops. Before the storm hits the beautiful sunset is used to emphasise an ending of life before. The horror of the storm that blows up to bring Dracula ashore is shocking even to Whitby, which is used to dramatic storms. After the storm is over, Mina notes that 'big, grim-looking waves' remain in the harbour as calm has not truly returned once Dracula is there.
The weather then remains grey, until the quest to destroy Dracula. The weather is good when the quest is going well, such as when waiting in the park in Piccadilly for the locksmith, and earlier after Lucy's final death. The weather is then neutrally grey until the final snowstorm. This storm is like a protagonist fighting against the heroes until they finally destroy Dracula, allowing the metaphorical sun to return.
Personal knowledge - Whitby faces north, and the next land one hits would be the Arctic so it is particularly subject to intense storms.
The Sublime. The Enlightenment [Year 1 core course] / University of Essex (1996/97)
Dracula / Bram Stoker [University of Adelaide etext)
"Sublime" in Glossary of Literary Gothic Terms / Douglass H. Thomson
Edited and updated essay:
To be added later