The meaning of both sentences is similar but there is a slight difference in meaning due to the tense used.
'I worry about you' is in the simple present tense which is used for regular and repetitive actions. Hence, this sentence means 'I worry about you' on a regular basis. For eg. 'You live alone in a big house. I worry about you.'
'I am worried about you' would more often be used when you are not talking about a regular action. For eg. 'You look thin today. I am worried about you.'