Yes, there is a difference. To put it in construction terms, a business plan is the scaffolding, while the business model is the foundation and building materials.
Far too many business schools focus on teaching their students to create business plans, without ever teaching them the art of creating business models. The result of this flawed approach is what we see today, a host of businesses operating under an "if we build it they will come mentality".
Entrepreneurs who think, "Hey, I have a great idea! I'm just going to get this thing built and then see if I can figure out later how to get people to buy it" are what I refer to as business plan entrepreneurs. Think Twitter.
On the other hand, business model entrepreneurs might think, "Wow, there seems to be a huge potential market for my idea, and it would definitely solve a need/pain...I think I'll do some testing and market research to see if this is really the case. Then, if the market really is there, I'll figure out how to build this and make it cash flow profitable right from the start". Think 37signals.
Do you see the incredible divide between these two approaches?
Think of it from the perspective of an investor. Angel investors and venture capitalists expect there to be a large measure of risk in their investments. They expect many of the companies they invest in to fail, or at best break even. Why? Because so many of them were trained in the art of the business plan, as were the people they are investing in.
Business plan entrepreneurs will almost certainly burn a TON of money, possibly for years, while they try to figure out how to monetize their company without straying from their carefully prepared plan. They are so focused on executing the original plan that they far too often fail to adapt and pivot as needed.
Business model entrepreneurs are the complete opposite. They think about money right from the start. They approach capital with a bootstrapping mentality, spending when it's needed, but never just because the money is available to be spent. Instead of a plan, they have a flexible outline, bullet points instead of paragraphs, and at least 3 or 4 different monetization options in mind.
So please, pretty pretty please with a cherry on top, don't be a business plan entrepreneur! If you are a startup, the only metric that matters is cash flow, and don't ever forget that. Don't start building anything until you've sat down and figured out AT LEAST 2 different ways to monetize your idea (and preferably 3 or 4).
And then, once you've figured out how to sell it, build it and test your ideas. Start selling. You'll never forget the feeling the first time someone buys your product.
I promise you, if your idea is good, addresses a need, has a large potential market, and you've already come up with monetization options, you will get the money you need when you approach investors.
If you have already started selling product, proving that your model is viable, scalable and repeatable, you'll get more money from investors than you ever dreamed possible, because you've alleviated a great deal of the risk for those investors.
So come up with a business model, not a business plan. Once you've proven your business model, feel free to plan to your heart's content.