For example, I've come across two different words, 'futuro' and 'estonteco'. According to several dictionaries 'futuro' relates to the future tense, whereas 'estonteco' refers to a future time. How are these different? How would you use these in a sentence?
How are these different?
As you noticed yourself, "futuro" usually* refers to the future tense and is thus a term to talk about grammar, while "estonteco" usually* refers to future time, i.e. to what is yet to come (whether talked about in the future tense in a particular language or not).
What is the correct word for “Future”?
Because the English word "future" can refer to at least two distinct (but related) concepts, this depends on which one of these are intended. If you have to translate the word in a text that isn't by yourself, you'll often have to infer the intended meaning from context.
How would you use these in a sentence?
Here's a paragraph from the lernu.net grammar pages (section on future tense) that uses both to explain how Esperanto's future tense (futuro) is used to refer to "the future" (future time(s); estonteco):
Futura verbo, verbo kun OS-finaĵo, montras, ke la ago aŭ stato ankoraŭ ne komenciĝis en la momento de parolado. Kompreneble la estonteco ĉiam estas necerta, sed futuro montras, ke la parolanto pensas, ke la afero vere okazos[.]
The same in English:
A future verb, a verb with an OS-ending, shows that the action or state has not yet begun at the moment of speaking. Of course the future is always uncertain, but the future shows that the speaker thinks that the matter will really occur[.]
Unfortunately, in English "future" is used a a synonym for both, "future tense" and "future time" without explicitly writing "tense" or "time", so the different meanings have to be inferred from context there. But Esperanto isn't the only language that distinguishes these two concepts by using different terms. Here's the German version:
Ein Futur-Verb, also ein Verb mit der Endung OS, gibt an, dass die Handlung oder der Zustand noch nicht begonnen hat im Moment der Aussage. Natürlich ist die Zukunft stets ungewiss, das Futur zeigt aber an, dass der Sprecher denkt, dass die Sache tatsächlich geschehen wird[.]
If you know other languages, try switching to them on that lernu.net page to see what terms that paragraph uses in them.
Of course it might be that some languages (like English) also don't distinguish between the two concepts with different words or that in some languages the paragraph is phrased differently (e.g. in French where the second sentence begins with "Malgré l'élément d'incertitude qu'il comporte, [...]", which translates verbatim to "Despite the element of uncertainty that it entails, [...]").
The fine print
* All that said, it seems that "futuro" and "estonteco" can be used synonymously in Esperanto.
- Λ Tenso, uzata, kiam oni esprimiĝas pli frue, ol la aludata procezo. ☞ -os.
- La estonteco.
- GRA Verba tempo esprimanta estontan agon aŭ staton. →perfekto, preterito, prezenco
- (malofte) =estonteco mi sciis, ke mi nun forludis jam mian futuron.
(bold emphasis mine)
And vice versa, "estonteca" can refer to the grammatical tense, at least according to PIV (see third meaning given):
- La tempo, kiun ni konscias kiel venontan: [...].
- La estonta sorto: [...]
- Λ Futuro.
I'd assume though that using the two words "swapped" like that is not only much less common, but would require a context that makes sufficiently clear which meaning is intended, at least if the distinction matters. While not wrong, I'd recommend to avoid using "futuro" for time and "estonteco" for grammatical tense, unless you have a good reason.