Keep with next means keep this current paragraph on the same page as the start of the following paragraph.
Keep lines together means keep all lines in the current paragraph on the same page.
You want to be judicious in your use of these. If you have both set for every paragraph, you will get bizarre page breaks because Word can't keep an entire document on a single page. That is what setting both does.
See:
Did you take a look at the sample I attached?
Style for following paragraph will almost never be correct 100% of the time. When it is incorrect, apply the correct style yourself. If it is correct most of the time, it is a time saver.
The paragraph formatting dialog box has two tabs. Most of the things we have been discussing are on the second tab.