First of all, when evaluating the sharpness of an image, it has to be done at 1:1 view - any other view will be inaccurate and misleading because the image has been scaled and resampled.
In addition, Library and Develop use different previews (the Develop previews are created in real time), so there may be some small differences between them. If there is a big difference in sharpness in Fit view, it's usually caused by excessive sharpening.
Even then, the image should display identically in Library and Develop at 1:1.
Please post a screenshot of the entire Lightroom window with the image viewed at 1:1, and the Basic and Detail panels visible.
As for the exported image, you will need to view that at 1:1 (100%) magnification as well to get a true impression of sharpness.
If the image has been properly sharpened in the Develop module (capture sharpening), using Output sharpening set to Medium (for screen or print) should produce good results.