I'm having a hard time understanding when to use 了 as a modal particle (instead of as an aspect particle).
For example, what is the difference between the following two sentences.
What are a few other examples of this usage of 了?
The two sentences in your example are the same. Native speakers usually use the latter, but if you use the former sentence no one will notice.
I'm not quite sure what a modal particle is so I will write some sentences ending in 了.Unlike the example, these 了 can't be omitted.
她八岁了。 She's eight years old now.
我们快要到家了。 We're almost home.
夏天快到了。 Summer is coming.
他已经走了。 He's already left.
学中文太容易了。 It's so easy to learn Chinese!
Usage of 了 (le)：
A. auxiliary word (助词)
A.1 used after verb or adj to indicate completion. This usage carries the same sense of "Perfect Tenses" in English.
A.2 used at the end of a sentence, or in the middle of sentence but right before a pause(usually a comma), to indicate current situation changed or new situation coming up.
B. modal word (语气词)
B.1 to indicate the mood of future certainty.
B.2 to indicate speeding up or stopping.
B.3 to indicate interjection.
C. Language Sense (语感)
Each language has its own language sense. By language sense I mean something about language that you can feel if it is good or not. Language sense helps us to know the way how to speak a language in a smooth and natural way, as native speakers do.
For example, in Chinese, we can say:
But translated into English, it will become something like
In this sentence, the 2nd "the" is needed. Why? Language sense. But in Chinese, there is no need of such article in front of the 2nd 桌子, and we all know that both the 1st and 2nd 桌子 refer to the same table.
So why are the 2 "his's" needed? Language sense. Someone may argue that without the 2 "his's", we don't know whose lunch he ate, and whose leg he broke. True, but again why do we need those specifications in English? Language sense.
In Chinese we will say:
See, here we don't say 吃过*他的*午饭, nor 摔断了*他的*腿, why? Language sense. If you say 吃过他的午饭后，他向屋外跑去，可却绊在门槛上摔断了他的腿, then it is not wrong, but it sounds very bad, as this is not the natural way native speakers speak.
Chinese is a contextual language, which means a lot info of sentences are hidden in the context. Chinese language also relies heavily on default info (缺省信息 or 默认信息). If no specific info expressed in sentences, then we just interpret them in the normal situation, by drawing info from the same sentence or from context.
So in some cases, need of 了 is just language sense. You need to get yourself used to it and familiar with it.