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Use your personal picture in your Twitter profile. This picture shows the face behind the brand. Use your logo as your Twitter icon only if you have a business account (for example, @Brandswag).
People want to know faces, not logos, because it makes you seem a little more human on a technology-driven tool. If you don't have a decent picture that you can use as a profile picture, consider having a professional (or, at least, a good amateur) photographer take your photo.