I'm not an expert on the difference between distros, but if you still want to mess around with Mandrake (or is it Mandriva now?), you could check to see if any processes are running that are hogging your resources. Try typing 'top' in a console and it will show you a list of processes that are running, their use of resources, and general system statistics. Type 'h' to get a list of commands to manipulate the program. See if you're running any resource hogs.
Also, it might be a hardware compatibility issue. What is the exact make and model of your laptop?
Other than that, it seems like you would have to do some serious tweaking, so you would probably do well to install another distro. For the record, I am also curious about what really makes the difference between distros. If you added apt-get to Slackware, why wouldn't it turn into Debian?