A website should have a unified graphic theme, as well as a tone and general feel. If a "professional feel" is what you're going for—if you're trying to sell a professional product or service, like web design—then aim for a professional look and design, and write the content with a professional tone. It's hard to describe because the concepts are often abstract, but a good designer can come up with a look and feel for any theme.
Besides helping the user subconsciously identify your website with a general category (professional, playful, serious, etc.), it's important to have a unified theme so that the user knows when they're actually at your website. That is, from a cognitive psychological view: if a user clicks a link, he should know whether he is still at your website or if the link has taken him to an external site.