The two modules use a different color space and the only accurate preview is within Develop module at 1:1 or greater. So what you report is to be expected. That said, the differences you show are pretty large.
Do you calibrate and profile your display and if so, does the software support, create V4 ICC profiles? LR doesn't like em, set the software for V2.
I'd heard that the color space issue was more of an effect of the noise reduction/sharpening of images, not in how color balance is massively distorted.
My monitors are towards the cheap end, an Acer and AOC, and have no clue for how to calibrate or maximize them properly.
Ideally you should calibrate and profile the monitors, as others have said.
However, here's a quick fix that will at least confirm what the problem is. I assume you're using Windows; go to Control Panel, Color Management.
First note what the default is now, if any (so you can go back there if needed):
Then select "sRGB..." and click "Set as default":
If you can't see sRGB in the list, click "Add...", and find it there. If you have two monitors, this can be set for each independently.
This will tell Windows and Lightroom that the monitor behaves as though it has sRGB colour space.
This won't be right, and it is always better to calibrate/profile properly, but for most monitors it will be very very roughly right.
More to the point, it may make the Library and Develop renderings look more similar.
If it makes things better, then your problem was no monitor profile, or a faulty profile.
--
Simon