the difference is specialization.
MS in Finance implies your future career is going to be in Finance. it is a very narrow / concentrated program with lots of numbers crunching. in addition, it is not always available as a part-time option for working professionals. if you are absolutely in love with Finance and you cannot imagine a change down the road - go for it.
MBA implies your future career is going to be in management (not necesserily Finance). it is a broad / generalist degree and you can pick your own concentration if you want, but you will have to take Finance / Accounting / Management / Marketing, etc to get a better perspective. there are lots of part-time programs. if you want more flexibility in your career down the road, and if you want to become a manager or may be even start your own business - go for this option instead.
I just finished my MBA and I can tell you that roughly 1/2 of the people I went to school with were engineers who are looking to develop expertise in managing business. this is pretty common in general and there is a certain logic to it. my personal advice would be to get into the best scool you can, get an MBA with concentration in Finance. you will thank me later
. in the end of the day, one of the main reasons why you would get a masters in business is to network with future leaders.
that said, please do yourself a favor a get some work experience before you decide on your MBA. highly recommend at least 4-5 years of relevant professional experience, otherwise you simply won't get the same bang for your buck and won't bring anything good to the table on team project work. I had to deal with people who have very limited work experience and they have been a drag on everyone.