Let's put it this way. Assume that you could match the IPs up and conclusively show that it is the same laptop. Then what? My point being that regardless, this is a matter for the police, and your efforts are more likely to hurt your chances of ever getting the missing laptop back.

And in reality, the odds of being able to conclusively prove something via IP address is very difficult. Something harder to alter, like the system's serial number, is a MUCH better bit of evidence. The IP address given by these tracking programs is just to help you find the physical location of the laptop, not necessarily prove that it is the same laptop.

Call the cops, be sure to mention that you have this tracking program installed, so they can start calling local ISPs to find out if that IP address is within their block of addresses. Then the cops can get a warrant to compel the ISP to cough up the address of the customer, get a search warrant, and potentially retrieve your stolen property. Then they can arrest the person at the same time, and you have the police report which will likely make any civil lawsuit you want to bring against the person an open and shut case. But the important first step here is bringing the police into things. A lot of police stations don't really have anyone necessarily all that tech savvy, so feel free to offer advice like the above, but make sure it's the cops that are the ones who are doing the actual work. Otherwise, you could very easily find yourself in some legal trouble of your own.