First, you should only do Cover Art maintenance on the MC Server PC, and not on the MC Client PC. If you do it on the Client, the Cover Art will be stored locally, and not be available to the Server.
Second, if there is already Cover Art imbedded in a file, common in audio files, I believe that will take precedence over Cover Art stored in the same folder as the file. I think this is particularly true when viewing the media on JRemote, but I'm not sure about that as I don't use it. If you have newer Cover Art that you want to use, you should remove the older imbedded Cover Art, and then re-add your new Cover Art.
To check if there is imbedded Cover Art, select the media file using MC Standard View, then look at the tag [Image File]. If there is embedded Cover Art it will say "Inside File". Otherwise it will say "Folder.jpg" or a file name, or a file with a path, depending on your settings when the record was created.
To remove the embedded Cover Art, select the media file(s) using MC Standard View, right click on them, select Cover Art>Remove Cover Art. Make sure you have copies of the Cover Art you want to keep though, as I think this removes all Cover Art, including files in the folder with the media file.
Tagging from the Client works fine, and is safe to do. Finding and assigning Cover Art doesn't work properly, and neither does the "Rename, Move, & Copy Files..." function, although access to the latter from a Client has been removed in recent MC versions, so it can no longer be used by mistake. Also editing Views from a Client doesn't work properly. Basically, other than tagging, and Playlists/Smartlists, it is best to do maintenance directly on the server. If you search you may find discussions on what works properly on a client, and what doesn't.