In 『よつばと!』, よつば says 「ここ家がいっぱいある」 to which her father replies 「そうだろう」.
What's the difference between そうです and そうでしょう? I know when we use そうです but I can't figure out the other. I know ましょう is used to suggest stuff, but I don't know how it fits with そうでしょう.
～だろう/でしょう in this context does not represent the uncertainty or guessing of the father. This actually is a strong affirmation of the Yotsuba's observation. I would translate this そうだろう as "Indeed," "Exactly," "Just as you say," or simply "Yup." I feel translating this as something like "So it seems" or "I think so too" is too weak.
If Yotsuba's father had said そのようだ or そうみたいだ, then it would have been "So it seems." Or if he had added extra words and said 多分そうだろう or おそらくそうだろう, then of course it would have meant "Probably it is so."
Then, what's the difference between "そうだろう/そうでしょう" and "そうだ/そうです" in this context, when both versions represent almost the same level of certainty?
Customer: あら、この財布は安いわね。
Shop staff: そうでしょう！Customer: あら、この財布は安いわね。
Shop staff: そうです！
In both examples, the staff strongly assures that the wallet is a good deal. But I feel the shop staff is a bit more friendly in the first example. Maybe it's related to euphemism?
よつば: ここ家がいっぱいある！
父: そうだろう。
Hmm. I would say this そうだろう is a bit more friendly version of "そうだ", but a bit more dignified/paternal version of "そうだね／そうだな". Anyway, the difference is not in the level of certainty.
Without context, I translate 「そうです」to "It is." and「そうでしょう」to "It is, isn't it?". But「そうでしょう」can be taken in different meanings depending on the situation, relationship between the speaker and listener(s) and intonation. I think of three situations to use 「そうでしょう」.
Authorization to confession or apology.
A: 「申し訳ありません。どういう結果を引き起こすか、慎重に考えてから行動すべきでした。」
B: 「そうでしょう。今後二度とこのような事の無い様に注意してください。」
Persuasion.
C: 「これは我々が責任を取るべき問題だとおっしゃっているのですか。」
D: 「そうでしょう。それが唯一の解決策ですよ。」
Agreement without being so sure.
E:「この書類の提出期限は月末でしたよね。」
F:「そうでしょう。担当部署に確認してもらえますか。」
The main semantic distinction that I'm aware of is that they represent different levels of certainty or previous awareness.
As a really simple example:
Person 1: それは ねこ ですか [is that a cat?]
Person 2: そう です [it is]
As opposed to:
Person 1: それは ねこ です [that is a cat]
Person 2: そう でしょう [yes, so it seems]
Apart from that でしょう is often used as an alternative to です when the speaker wants to appear less forceful in what they're saying. It suggests the possibility of something being the case rather than explicitly stating that it is.
To keep it simple, let's assume that you're only talking about そうです and そうでしょう as "complete" statements, i.e. when responding to something, and let's ignore the そう part for a while to better focus on the main point here, which is the difference between です and でしょう.
While です and でしょう are both used to express a certain level of politeness, and usually considered to bear the function of copula, です in most contexts is the lexicographical equivalent of a neutral "be" or "is" in English.
でしょう on the other hand, is in one word roughly the equivalent of "seems". Depending on the situation, other frequently used translations include "I think", "I guess", "I wonder" or even "I hope". In many cases でしょう can be translated as "don't you agree?" or "that's what I thought!", much like the particle ね.
Translations of phrases including でしょう would usually include a "be" or "is", but by itself it often matches phrases like "yeah, I guess", where "be" or "is" may be omitted even in English.
It's thus necessary understand that context in which でしょう is used, to choose a suitable translation for a given situation. Re-inserting そう into the equation; While そうです means "it is so", そうでしょう should be interpreted as "so it seems", or depending on the context, something like "isn't it so?" or "that's right!" etcetera.
Side-note: While the phrase そうだ is often used as an exclamation upon realizing or remembering something, I've never heard anyone use そうです in the same manner, and I don't think that would fly very well. In other words, even though "that's right!" can be a viable translation also of そうです, it's not so in the context of "talking to yourself".
In more complex sentences like 雨{あめ}が降{ふ}りそうです, the そう construct has a somewhat different function as compared to the above, and is used to modify the main clause rather than to give feedback on someone else's statement. The difference between です and でしょう though is still the same.
Edit: The OP changed the question from being general to adding a specific context. In this named context, そうだろう should be interpreted as a confirmation of the observation made by やつば, much as suggested by @naruto, who does a good job trying to get at the difference between そうだろう (そうでしょう) and some neutral confirmation (そうだ, そのとおりだ etcetera).
Moreover, I would suggest that the reason why the context of "adult" vs. "child" makes the phrasing from the OP's example natural is that (in combination with the informality of だろう) it also hints on the nature of the relationship, as in the adult being more experienced and knowing things that the child has yet to discover.
This is because in contrast to a neutral confirmation, そうだろう in this case implies that the speaker either has already made the same observation (obviously without the need for confirmation from the child), already knew that "there are a lot of houses here", or possibly just feels that it's an obvious conclusion based on some other fact.
To me, understanding the relationship between speaker and listener is necessary to comprehend how an often uncertain or seeking-confirmation-like phrase as だろう suddenly turns into something really quite the opposite. (Of course, the same phenomenon can be observed not only in Japanese, but in other languages as well, including English.)
