The primary difference between the two is the time value of money. ROI does not consider this, its equation can be easily broken down into (Return-Investment)/Investment. It's a simple calculation to find how many dollars of return was produced by each dollar of revenue.
To understand what IRR is, you can ask yourself the question, would you rather receive $100 now or a year from now? Assuming no outside conditions, everyone would say now is preferred. Why? Because money is worth more to us now then in an uncertain future. Since accounting is all about putting numbers on things like this, we calculate an interest rate to determine how much the money today is worth today rather than later. This combined with the cash flows from an investment, creates the IRR. The IRR is the value of the time we put on those future cash flows in the investment.
In conclusion, the main way you can look at it, is that the ROI is the rate that the investment money makes us return money. The IRR is the value we place on the time to get that return money in the future.