The only real issue that you will have is that the default filesystem for the two operating systems is different, and neither system will read the others disks, so plugging the disk from one computer into the other is not possible. You would need to have another drive, which you would need to format as FAT32, which is readable by both systems. You can then copy both your .LRCAT catalogue file, and of course all of your images to the intermediate drive for transfer. You will need to keep the folder structure for the images identical across the two machines, during the transfer to make having LR find your images again an easy job. That deals with your images and catalogue files, but there are other files that you will need to move, all of your import, export, develop, brush, filter, print, and every other sort of preset has an individual file associated with it, and you will also need to find all of them and move them to the new computer, where you may very well have to manually enable them for use on the new system, the files are cross platform, but you have to make the LR install use them. LR is pretty good, because although there are lots of different file types for this, with a variety of odd extensions, they are all as far as I know actually all simple ASCII text files. It is possible to open all of the files in a simple text editor and read the contents. Actually it is even possible to use a simple text editor to edit your RAW files, using ACR and .xmp sidecar files at least, it is how Dehaze, normally only available in the full version of ACR is able to be implemented in PS Elements, which doesn't get that feature, since the conversion engine is identical under the hood. You can add the local brushes and filters that way too, but programming the brushes by text is much harder to do for complex shapes. So don't forget all your ancillary files too. I got caught out with this, since I wasn't specifically backing them up, and when my computer went down I lost all my saved presets, which I am having to recreate as I go along. Most of them have not been to problematical, but I did have lots and lots of printing presets, and saved prints, that I had created which I also lost, but I do have the smart collections that they created, just not the print settings associated with them. So it is important to ensure that you also back these files up, and where necessary transfer them across. I was caught out by this, since I had thought presets were part of the catalogue and so backed up with it. I did not bother backing up the program files, since I could easily reinstall the program by downloading the latest version of it from the Adobe website.
Alan