Suggested answer:
(1) Examples of the searches remember *ing and remember to:
I remember thinking I heard something.
Remember to wear your whites, it's hot down there.

(2) Although both search strings provide examples of remember not followed by a verb, it is remember *ing that provides most examples, e.g.:
"I think I do remember something about that," the librarian replied.

(One rare example of remember to not followed by a verb: They hired this ancient-looking guy, I can remember to this day.)

(3) When remember is followed by a verb ending in -ing you have a memory of having done something, i.e. you remember doing something after you have done it, e.g. I remember asking if it was okay... could be paraphrased as: 'I have a memory of asking you something', or 'I asked you something and now I remember doing so'.

When remember is followed by the to-infinitive you remember to do something before you do it, e.g. Will you remember to give them that message? can be paraphrased as: Will you remember that you have to give them the message so that you can do it?

So, the main difference between the two can be said to be that remember followed by a verb ending in -ing looks backward in time, while remember followed by the to-infinitive looks forward in time.

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Read about verb + -ing or to-infinitive here and here
Terminology: -ing participle clause, to-infinitive