A single duplicate target cannot possibly exist for this type of question because whether an infinitival or gerund-participial is used depends on
- the type of verb, or the individual verb licensing the above forms, and
- sometimes, the complement itself.
First of all, this is not about the specific forms of the verb functioning as complement – in this case jump (and the forms (to) jump and jumping) because most verbs, with some exceptions, commonly occur in both of these forms when functioning as complement.
The crux of the matter is which verb or verbal idiom is followed by the above-mentioned forms. More precisely, the difference in meaning depends on the verb which licenses a (bare or to-) infinitival and/or gerund-participial complement, rather than the verb used as the complement itself (with some exceptions).
A very common example is the idiom go on. Here's the relevant entry from Practical English Usage by M. Swan:
Go on + -ing form means 'continue'.
She went on talking about her illness until we all went to sleep.
Go on + infinitive refers to a change of activity.
She stopped talking about that and went on to describe her other problems.
Note how the author doesn't place any restrictions on the verbs that can be used as complements. A more important thing to observe here is that there's a separate entry for the idiom go on. There are also separate entries for verbs regret, remember, advise, see, stop, try, and a couple others, as well as adjectives, where the difference in meaning between infinitival and gerund-participial complements is explained.
Let's look at another entry, simply to make clear the fact that we cannot provide the same explanation when it comes to the difference in the type of complement:
see, watch and hear
After these verbs, the difference between object + -ing form and object + infinitive is like the difference between progressive and simple tenses. With -ing forms the verbs suggest that one pays attention to events or actions that are already going on; infinitives usually refer to complete events/actions which are seen/heard from beginning to end.
There probably† exists a relationship between these two entries. After all, we are dealing with the same type of constructions, and they're bound to be related semantically, but it's dubious whether you can actually generalize this difference in meaning to all verbs and adjectives and idioms to which the difference applies and is appreciable.
As I mentioned, there are also some exceptions with respect to the verbs used as complements. The entry for the verbs begin and start contains the following:
begin and start
Infinitives are also preferred with understand, realise and know.
I slowly began to understand how she felt. (NOT ...
He started to realise that if you wanted to eat you had to work. (NOT ...
† The Cambridge Grammar of the English Language (2002), commonly referred to as CGEL, written by Huddleston et al., on pp.1240–1 says:
Which form-type(s) a particular catenative selects must be specified lexically for that verb: we cannot assign distinct meanings to the form-types and treat the selection as semantically determined. On the other hand, the selection is not random: verbs with similar meanings tend to select the same form-types, and where a verb allows both major form-types we very often find a difference in meaning that is at least partly motivated by their general characteristics.
We have noted (§1.4) that infinitival to derives historically from the preposition to and that while it has quite clearly undergone a syntactic change such that in this use it is no longer a preposition, certain aspects of its infinitival subordinator use reflect its origin. Prepositional to is characteristically associated with a goal, and a metaphorical association between to-infinitivals and goals is to be found in the fact that they commonly involve temporal projection into the future, as with the complements of ask, choose, consent, hesitate, order, persuade, promise, resolve, strive, tell, threaten, and countless other catenatives. Linked with this is the modal feature of potentiality. The gerund-participial, by contrast, is commonly associated with what is current and actual, as in They enjoy walking, She finished working, He practised speaking with an American accent, and it is plausible to see this as connected with the nominal source of most gerund-participial complements. But it must be emphasised that we are talking here of historically motivated tendencies and associations, not constant elements of meaning.
CGEL then goes on to list 6+ categories (one of which is titled Some individual cases) of verbs over the next 4+ pages with respect to the potential difference in meaning between the usages of to-infinitivals and gerund-participials.