Liberty profile is light weight version of WebSphere Application Server(WAS). All the versions like base, core and Network deployment(ND) supports basic application execution, they vary with other advanced options like base and core will not support cluster deployment, cluster can only be supported in network deployment(ND).
Regarding Fix Pack installation: Liberty profile is shipped with WAS or alone(only liberty profile without WAS). You can install only the same flavor of fix pack, which WebSphere application server is installed. For eg. If WAS is base then only base fix pack can be applied. If WAS is ND, then ND fix pack can be applied. Same applies for only Liberty Profile as well. You have to install same flavor of fix pack for Liberty profile, which you have installed. Fix Pack are like patches on major release. Fix pack can only be applied on top of Major release for WAS/Liberty profile.