To answer your question:
Onyomi is more formal and mature.
There are nuances with a lot of the related words. So I would recommend you to use Keigo or polite です・ます形 with Kunyomi words until you have a better grasp of the situations that they are acceptable and not acceptable to be used in.
I like to compare the difference between the two types of words with a thin lined circle and a thick black line.
I also think that @Yoichi said is true where there are no rules to judge the usage between the two.
Kunyomi
In spoken Japanese; in casual and day to day situations the Japanese Kunyomi words are used and have a softer, rounder, feeling like that of a circle.
Kunyomi words, like mentioned above, have a softer feeling and are often used in place of Onyomi in situations where a certain action or thing is not recommended, frowned upon but not against the law, or where Japanese want to show 遠{えん}慮{りょ}/consideration by being soft about their wording.
An example using question's provided words would be that of signs.
A lot of places close to schools or on train platforms will have signs and or posters with the words あぶない!・危ない! in either Hiragana or Kanji with 送{おく}り仮{が}名{な}Okurigana. This is to let people know its dangerous and to be careful of kids crossing the street or to suggest standing back when trains are arriving at the station. You could think of it as a friendly reminder.
Contrarily, if you were driving on the street you might also see a round sign with 危険物{きけんぶつ}or a car driving with 高圧{こうあつ}ガス危険 on it. Clearly stating dangerous object/thing or "Beware, High-pressure gas". Albeit, you will also find the Onyomi words written in Katakana or Hiragana and this is usually written so children who can't read the Kanji can still understand the signs message therefore the danger.
Also, you will probably never hear a mother yell 「危険だよ!」 at her son playing near the street, she will most always say 「危ない」.
Other examples include posters regarding manners for the public where using Kunyomi is preferred because it is less intrusive and lighter which you can often find in subway stations.
Onyomi
On the other hand; business situations, formal occasions, Japanese literature, and non-spoken communication like above mentioned examples often use the Onyomi. This is because Onyomi words are deep in meaning, a lot stiffer/stricter than Kunyomi and have weight. Due to this fact they get straight to the point like a big black line. (Circle and line)
Uses:
-In Japanese literature reading is faster when you can describe something in fewer characters. Of course there are always exceptions to the rule but a lot of Onyomi words in books are also classified as 書{か}き言{こと}葉{ば}. Using written words or 書き言葉 sometimes throws people off with the sudden change to a super formal or polite word(Onyomi word). You might get a few weird responses and or looks.
Ex: If you were talking to a friend and said: 今日の仕事は終わったよ。- Would be
considered normal, friendly and casual.
今日の仕事は終了しました。- Is casual but then you are putting a more formal Onyomi
word at the end. You might not get the same response.
本日の作業は終了しました。This would be a business or formal sentence with the same
meaning.
-From my experience the seriousness of Onyomi words can carry a lot of weight. In this sense using them in daily speech can be a little over bearing. However, Onyomi words are in most cases more specific and deep in meaning and I think using them where they are needed is your best option. For example, if you are talking about something more specific, talking about research or something you you know a lot about.
Notes:
-Sometimes you run across older folk that speak very traditionally and will use a lot of Onyomi words. This doesn't mean that you should.
-This also applies with parents and a lot of the time it is just the way they were raised more so than a choice. Some of my friends parents will speak in Keigo to everyone including their own children. Getting used to hearing this can sometimes take a while and can be sort of scary at times.