These two descriptions are actually used interchangeably quite often. They are extremely similar. In the vast majority of situations, I wouldn't worry about the distinction between these ideas.
If you ever found yourself in a situation where you absolutely had to differentiate these, you can, but I'd honestly be surprised to see the LSAT require this level of specificity. (If there's a question that does this, I'd love to see it - feel free to post it or PM me!) If I saw these in two answer choices, I'd expect there to be some *other* critical difference between them that would definitively kill one answer.
That being said, I would expect "using examples" to describe an argument that is attempting to prove a general concept, and is using specific examples to back it up.
All boys must like sports. After all, Joey likes sports. So do Jack and Bob.
Argument by analogy, on the other hand, is more likely to use specific examples to support a conclusion about *another specific example* - the author would be drawing an analogy between the premise scenario and the conclusion scenario.
Jackie must be late to class. After all, she was running across campus, and every time I see Billy running across campus, he is late to class.
One reason you might see the terms used interchangeably for the second version is that it's easy to think of this argument as having an implied intermediary step: Every time Billy is running, he's late; therefore if someone is running, they are late; therefore if Jackie is running, she's late.
That implied general principle in the middle is being supported by the specific example about Billy.
Don't get too caught up trying to split hairs between these types of terms unless you have no other choice! (and if you think you have to split this kind of hair, there is probably something that you've overlooked!)
Does that help a bit?