Basically Islamic Finance falls into two categories: Banking and Investing.
With banking, most Islamic banks are basically charging interest in an indirect manner. They operate using Lease to Buy schemes or something similar. The bank buys the house and then you pay the bank each month in order to increase your share in the house, with an additional "rent" payment in order to live in the house.
The base payment you make is called Principle in traditional finance, and the rent is called interest. There is no difference. What they fail to recognize is the basic principle of lending in Islam: you should only lend people money out of goodwill and never to make money from it. A truly Islamically financed house would be financed in a sort of joint venture/partnership by the bank and homeowner. Both invest in the property and the venture pays off (or not) when it comes time to sell. But there would be no payments of any sort made that pays the bank over time. This kind of venture is very risky and hard to find investors for, and it can also be a very long term investment in the end (since the profit only comes when you sell the house and that could be 20 years down the road).
When it comes to Islamic investing, these investment companies are basically saying that it's permissible to invest in companies that have debt, so long as the debt does not equal more than 33% of their equity, it's OK. Imagine that; 34% debt to equity and it's haraam, but 32% debt to equity: halaal! Some companies also find it permissible to invest in stocks that hold a small percentage of alcohol, pork, gambling, etc related investments (some scholars call this "permissible impurity".
This is unacceptable in MY OPINION. Are we allowed to drink something if it contains 32% alcohol, or 32% pork, etc? No. So why should investments be any different?
Investing in and of itself is not haraam; it's perfectly OK to invest in something provided the investment itself is halal. But the way the company runs their business must also be halal. They should not be borrowing money from a bank or using any other means of interest to finance their operations. If they are, then you as an investor are sharing in those profits derived from interest.
But because virtually every company under the sun uses a bank to finance their business (even companies like Apple and Microsoft with huge amounts of cash use loans to finance their payroll each period), investing in today's stock market is basically out of the question in my opinion.
Please remember that I am not a scholar, but merely a student of knowledge. If one truly has a grasp on basic Islamic principles and financial principles, then one would probably arrive at the same conclusion I have.
Also, google "Let's Buy a House Islamically". It's a great lecture that really opened my eyes to this stuff. Remember, don't take my opinion as a fatwa or anything, I'm just expressing my view.