Iris:
I have significant experience both as a Tech Writer and a BA. There is a difficulty in answering your question: What, logically, a Tech Writer should be focused on and what he/she typically actually devotes most of his/her work energies to are most often two entirely different things. This is not a "knock" against Tech Writers; the exact same issue pertains to BAs, SA, QAs, etc., etc.
I will only address what should be done. The primary goal of a software Technical Writer should be to come up with a comprehensive, integrated understanding of the system's essential functions (i.e., business functions \ processes that exist, irregardless of any implementation consiserations) and - especially - how those functions interrelate (typically, the data flows between them). Such is also the primary goal of a BA, but typically, the BA works to a higher level of abstraction than a TW (i.e, the BA is often most focused on the "bigger picture", while the TW is more grounded in the details.)
Difference between a functional doc and a development doc? These mean whatever your organization says they mean. Simular to your question, recently on the Requirements Engineering listserv, someone asked "What is the difference between a business requirement and a functional requirement". This is a very basic question, and there are some very knowledgable people contributing to that listserv. However, after a couple of weeks of discussion, no agreement was reached. As a matter of fact, just yesterday, I reasked the question, and there still is no agreement: Alot of well thought out ideas - but no agreement.
You will find the same thing regarding your question about the difference between a functional doc (which can also be called a functional spec, functional requirements spec, a systems requirements spec, a systems functional requirements spec, etc, etc, etc) and a development doc (which can also be called a design spec, a functional spec, a technical spec, etc, etc, etc) (Do yourself a hugh favor: Don't listen to those who imply that there is some universally accepted definition for these, or other simular artifacts.)
Tony