Using Printique online and LR for best results?

As already mentioned this is due to Classic only syncing smart previews which means the cloud service can only create jpeg images from it at the smart preview resolution. The solution is to create the jpegs yourself in Classic and upload them directly to the print service or you must upload full raw images to the cloud, which can only be done using Lightroom Desktop which is a totally different program than Classic. It is basically a port of the mobile version of Lightroom to the desktop. It is possible to do your initial sync from CLassic and then open Lightroom Desktop (do not let it migrate your catalog!) and drag the raw files on it and it should place the raw files in the cloud and associate it with the already uploaded smart previews.

Do you perhaps have a copy of ON1 Photo RAW? included in that is ON1 Resize. I like using resize to create a photo with exactly the desired output dimensions. This way no upscaling occurs at the print provider. 

"But after selecting a test image it said that it coul donly print to good results, not best results. I am wondering why?"

Maybe you should ask Printique. Or maybe you should tell us more about what you are doing, like what size (in inches) are you printing and what other requirements do you have.

"...but are the ones stored in the cloud different and how do I control the image resolution on LR Online?"

If you are using Lightroom Classic (are you?) then full sized images are never stored in the cloud. You cannot control this, it is fixed. Smart previews are used in Lightroom mobile when you send them from Lightroom Classic.

If you are using Lightroom, then you should have full resolution images in the cloud.