The 4 types of human emotion, 喜怒哀乐， seem to be 3. What difference do you see between 欢喜 and 快乐？
喜 delight; pleasure
樂 happiness; joy
If you do not know the difference between 喜 and 樂, you can realize it from their antonyms.
It is pretty often to use two opposite words together in a Chinese phrase. For example, (天地)玄黃, (天地)良心, (陰陽)怪氣, (陰陽)交錯, (緩急)(輕重), (緩急)相濟.
It is the same case in 喜怒哀樂. There are two pairs of words here, which are 喜怒 and 哀樂. Since the question mentioned 喜 and 樂 of 喜怒哀樂, let's just focus on this phrase. The antonym of 喜 is 怒 (anger), and 哀 (sorrow) is the opposite of 樂.
I think you certainly know the difference between anger and sorrow, then you should know the difference between 喜 and 樂.
天命之謂性，率性之謂道，修道之謂教。道者也，不可須臾離也；可離，非道也。是故，君子戒慎乎其所不睹，恐懼乎其所不聞。莫見乎隱，莫顯乎微，故君子慎其獨也。 喜怒哀樂 之未發，謂之中；發而皆中節，謂之和。中也者，天下之大本也；和也者，天下之達道也。致中和，天地位焉，萬物育焉。
I'm a native speaker of Chinese. I would like to explain the difference between "喜" and "乐" in English, but it's only my own opinion and the usage of English words may not that accuracy.
The background you need to understand is how Chinese language user usually express their emotion. Most of time, we prefer not to show our emotion in a directly way. So, you can say a person is adultness with "喜怒不形于色"(you don't show joy or anger on your face).
In general, "喜" is a more inner emotion in one's mind like joy, while "乐" is the emotion you show to all people around you like happy.
If someone is "欢喜", he feels delight because something he looks for is achieved. It's an activity in his mind and he can choose to show it with smile or not.
If somesone is "快乐", it expresses that he is so happy that he can't hide it and he would like to show it out. It's a stronger emotion than "欢喜".
Also, "欢喜" is a formal word usually used to writting article, "快乐" is a casual word used in both oral and written Chinese.
If you just want to know the meaning of 喜, and 乐 in 喜怒哀乐，the root will tell you everything.
喜 ： the word original mean something you prefer/desire. The word then later extend to anything that brings joys.
乐/樂 : glyph shows a person handle some instruments. Meaning then extend to something that cheers you.
For example, listen to a joke will cheer you up。 For example
Both carry the different meaning. The first sentence showed the person is pleased. The second sentence indicated the person feel joyful.
OTH, 欢喜 and 快乐 are not appropriate to use to distinguish 喜 and 乐. Because the usage of 欢喜 evolved over time, it can mean cheer, joy, happiness, delightful.
You may read something like this 她很得祖父母的欢喜, here. You cannot translate it to "She is the favorite grand-daughter" without reading the context.
In addition, when related to Buddism context like 皆大欢喜, it means delightful or satisfied instead of happiness.