Having been a scientist for going on six years now, I have little awareness of how outsiders view science, or scientists themselves. The best glimpses I get are from movies and television, where scientists are dressed in white lab coats, and work in colorful rooms. The truth, while not quite diametrically opposed, is much less sexy.
Almost no one in the lab actually wears lab coats. In fact, my high school biology teacher once joked that if you ever see a lab wherein everyone is wearing lab coats and goggles, you should run because they are working with very dangerous things. Typically, scientists dress like computer programmers, viz. jeans and a t-shirt. If you are more senior, you might wear what my friend calls the "scientist's uniform" of khaki slacks and a blue button down.
In fact, the dress code of the lab is so casual that I am instantly suspicious of anyone who dresses too well. When I see someone in a blazer, I wonder if they have a job interview. Or someone wearing a white lab coat, makes me wonder why they are trying to appear to be working (lab coats are of course essential for many lab procedures, like dissections. My rule of thumb is that you should never wear a lab coat without gloves). My lab recently bought lab coats for a few people, and they now wear them when they are doing any work in the lab, not just the dangerous or dirty stuff. It irks me. To be fair, one's attitude can be changed by the clothes one wears, and I would endorse any action that makes one work more effectively. I would need to see the data, though, that shows lab coats make them more effective.
Medical doctors are some of the worst offenders in terms of using the white coat as a status symbol. For doctors, the coats are certainly necessary when working with patients that may bleed or drip muccus. But the doctors often do not disrobe outside the office, and wear their coats in the cafeteria or on the way to their car. Part of being a doctor is certainly to make people as comfortable as possible, and wearing a lab coat may inspire confidence in patients. Like scientists, though, when they are worn too often, I become suspicious that they are compensating. It doesn't seem very hygenic, either, to be wearing your dirty safety clothes in public, but then again, I'm not a doctor.