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For example, what is the difference between the following two sentences?

> 私はメアリーとトムに会った事実を知っている。  
>  私はメアリーとトムに会った事実を知っています。
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question
word-choice politeness
What is the difference between ている and ています?
I'm guessing that you learnt 知っています first? This is the polite way of talking
about 'knowing'. 知っている is the 'plain form' that you would use when talking to
people you know well. In non-past tense these verb forms are those you will
find in a dictionary.

いる (plain/dictionary form) --> います (polite form).

Although, now I think about it, if you are comfortable with 会った then maybe you
learnt the plain forms first? To make the polite form you change the final
vowel sound from う to い and add ます.

かく -> かきます

よむ -> よみます

However if the final sound end in いる or える then the る is simply replaced with
ます

たべる -> たべます

いる -> います

Note that there are quite a few exceptions to this latter rule that you will
have to learn as you come across them.
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answer