People use all sorts of different images for their main twitter profile picture that appears next to every tweet you post as well. Some people just use an image of themselves, others use images of their pets of children;many people of course change them regularly perhaps putting up funny pictures they come across. Many businesses and websites though use a logo from their site or another image related to what they do or perhaps even where they are based.
Each of these different picture types has an advantage except perhaps those that are totally unrelated. If you are using Twtitter purely socially and for fun then what you use isn't too important: a picture of yourself though helps your friends find you and make sure you are who they think you are.
If you use your Twitter for work or it is a specific work account then you must be much more careful what picture you use, some pictures could actually damage your brand others can boost it and make it easier to get followers.
If you are a consultant or a freelancer then you may well trade under your own name anyway, a picture of your face may be best, especially if you manage to look respectable and intelligent in it. Also even in business people may look for people they have met, for example at networking meetings, based on faces as well as names to make sure they get the right person. Pictures of things unrelated to work such as pets and your children are very unprofessional and make your account look like a social account more than a business account. Pictures that may be more related are your offices, especially if they are fairly impressive and at least show you aren't a back bedroom operation. You could also add a picture of the town or city where you work to highlight your location, this makes good sense if you have the name of where you are based in your business name such as 'Newtown Carpets' with a lovely scenic photo of Newtown's famous skyline.
Logos though are perhaps the very best and most professional option for your Twitter account, they make you look impressive if they are well designed and increase awareness of your brand identity. Whatever you do choose though try to keep it the same long term, it has been shown that people look out for their favorite tweeters mainly by the picture rather than the name.