I want to understand the difference between System and Liblouis Braille table particularly in relation to Braille Screen Input using Unified English Braille.
Liblouis is open-source software that is developed by a small group of volunteers. The system tables are developed Duxbury, which is a company that develops the most feature-rich braille translation software in the world. They each support languages that the other does not, but I've generally found the Duxbury braille tables to be of a higher quality. Since you're not using a language that there isn't a Duxbury braille table for, there isn't really a need to use Liblouis.
You use Liblouis tables when using BrailleBack on Android and with NVDA and JAWS (but only JAWS 17 and later). Duxbury tables are used on devices such as BrailleSense notetakers.
The differences I have seen have been in translation when using UEB as my Braille code of choice. There are some things Liblouis does better, and some that the system table does better. The system table does not seem to be able to handle fractions in UEB, e.g. two and three quarters, so if you're going to be typing something like that it's best to switch to Liblouis to do it. Liblouis for some reason doesn't seem to have the word 'can't' in its contracted form in the braille table. So if you type C, dot three, T, you get c't not can't. The system table does this correctly. In an ideal world both tables would be equally comprehensive, but at least it's very easy to switch between them.