はじめまして is less formal and as the direct translation suggests means "(nice to meet you for) the first time". Thus, はじめまして would only be appropriate in less formal setting and only for the first time you are introduced to someone. It is more "nice to meet you" than "how are you", though in English there are (from the point of few of other languages) strange greetings like "How do you do?" which is not not a question and does not expect you to start explaining how are you, instead you answer "How do you do?" In more formal setting you would probably find people say よろしくお願いします or even どうぞよろしくお願いします. Which both basically mean "please be nice/kind to me" if you look at the literal translation, but can mean "nice to meet you" when used as part of formal introduction, or as "let's have nice business, etc together" when said in connection of having come to agreement of some plan (for example to do business together). Thus よろしくお願いします is not limited to only first time encounter. That's the beauty and pain of Japanese language, that many things mean one thing or another things depending of the context.