Gerunds and to-infinitives are the non-finite verb phrases as verbal complements. In English, gerunds and to-infinitives as verbal complements show the temporal and aspectual relationship of the verb and complements. There is a list of verbs, assigning which verb needs to use for gerunds or infinitives.
“To” can link with goals, direction in to-infinitives. When it correlates with the verb like order, ask, promise, it indicates a future-projection. For example, “I promise to buy you a book.” involves “forward-looking” constructions (Conti, 2011). The verb like tend, plan and hope, its complement is projected forward in time, into the realm of the future, into "forward-looking" constructs. It can also be used to indicate the targets wanted to achieve. “I want to swim” shows the desire/target of the speaker.
On the other hand, gerunds is the nominalization of verbs. The “V+ing” form emphasises the event or activity that is ongoing and already existed. For example, “I enjoy watching movies.”, the object followed enjoy need to be a noun phrase. So action nominalization is needed to become watching.
As mentioned, the to-infinitive is future-projection. Therefore, when verbs like enjoy, keep, admit are used with to-infinitive, they are semantically incorrect since the events followed by these words supposingly ongoing and already exist.
However, some verbs can use gerunds and infinitives, it relies on context to choose which should be used. It is special for verbs such as stop, forget, start and like.
For example, “She stopped to greet us” and “She stopped greeting us.” The semantic meaning of the two sentences is different. The first one indicates the girl stop what she was doing at that moment and then greet us. For the second one, the action of greeting us has done a long time, and the girl stopped doing that to us now.
For remember, “She remembered telling you the story”, it involves premise that the activity of telling the story may be a common practice before, and she recalled the moment in the past. For another case, “She remembered to tell you the story.”. It has two actions of remember and tell you story, and there is sequence of the two-action happened. The girl remembered first, and then the second action is projected forward in time.
The usage highlights the characteristics of gerunds and infinitives, and shows the relationship of the verb and verbal complements.