It can help at some points in the process. Not the fact that you made contact has influence, but the fact that by getting in touch and asking appropriate questions can assist you in making your application more relevant to their needs.
It helps you understand what research groups there are there, what courses they teach, what are the gaps that they are trying to fill and so forth. Then when asked at interview, or when making a presentation a question like "How do you think you are a good fit for his department" you know what to say!
Yes: you can learn much of this by reading their public facing materials like web pages, research output and publicity and so on, but remember much of this is targeted at recruitment of students (undergraduate and postgraduate) and not on the hiring of new staff.
Some advertisements often state "for more information please contact [email protected]", and they really mean it.