There are quite a few future tense constructions in Korean, each with a slightly different nuance.

-겠-

I like to call this the "assertion future". This is the most general in terms of morphology. It can apply to first, second or third person, and can be used in any speech style. Like many other tenses, it isn't strictly "just future tense use". The breakdown below follows the use in Comprehensive Korean Grammar:

• Formal statements about the future

잠시 후 기차가 출발하겠습니다.

The train will be departing shortly.

In English this would be a will-future. Very common in announcements, especially for the immediate future. Naturally this is paired with -습니다 in the formal deferential speech level or the -다 ending in the plain non-polite level.

• 1st person definitive promises/intended actions

내일까지 하겠습니다.

I will do it by tomorrow / I promise it will be done by tomorrow

The promise is made in the instant of speaking, although the content of the promise may be an action that lasts further into the future. As with all these intention-type constructions in Korean, it refers to the first person. It is a "definitive, solemn and binding" in its tone. Again, it matches the definite quality of the English will-future.

However, when paired with the negative 못 it translates to not being able to do something, and this refers usually to the near future.

• Inferred statements about 2nd/3rd person

우와! 맛있겠다!

Wow that looks delicious!

So this really doesn't have much to do with the future tense constructions of English or other Indo-European languages. It just points out what something "is like" based on the speaker's inference, and often corresponds to English "must" or "I'll bet that...", with the speaker being fairly sure of the statement's truth (compared with other "inference" constructions in Korean like -는 것 같다 or -나 보다).

• Formal offers and requests

여기 앉으시겠어요?

Would you like to sit here?

Again with either formal deferential or non-formal polite endings, these requests are softened by the use of -겠= and often translate to English "would [you] like to... ?".

• Tentative 알겠다 / 모르겠다 This is what I'd call "hedging" whether the extent one knows or does not know; it actually makes it more tentative. There's a lot of discussion on this, but for native English speakers that "would" re-appears in one possible translation, "I wouldn't know."

-ㄹ게-

The morphology of this is restricted to action verbs (not descriptive verbs) and to the speech levels of 해체 "intimate" or "non-polite non-formal" and 해요체 "non-formal polite". It also takes as subject almost exclusively the 1st person.

It is used primarily for:

• 1st person definitive promises/intended actions

...just like -겠- above. But its nuance is much lighter: not only is the speech level less formal and less "solemn and bound", the actions to be taken in the promise are usually more directly controlled by the speaker, and the hearer is usually directly affected. The intention is usually accomplished quickly, often immediately...

A: 안 돼! 난 돈 없네!

B: 괜찮아. 형이 살게!

A: Oh no! I haven't got any money!

B: It's OK. Your bro'll buy it (= I'll buy it)

The Grammar refers to these as "spontaneously formed intentions" which is a particularly nice way to describe it. I feel it can be more contingent on others' reactions and replies.

It cannot take -다 (non-polite plain speech level) or -ㅂ니다 (formal polite deferential speech level), hence it is replaced with its more appropriately "solemn" relative above. So for example 먼저 갈게(요) to colleagues but 먼저 가겠습니다 to senior managers (should they let you leave first!!!).

Hence -ㄹ게- would be quite common in everyday conversation because of its usage, but -겠- has its place too and has a greater range of meanings anyway. To this, there's also -ㄹ 것- and -려고 하다 as well, for other types of futurity and intention.

For further reference: TTMIK Grammar Level 8 Lesson 15, and How to Study Korean Lesson 63