C-3PO has a very emotional and extroverted personality. He seems to prefer Galactic Basic because it seems to be a good language to express his emotions.
But considering that R2-D2 is designed for much more technical tasks, one can assume that his beeping language is designed primarily for expressing technical concepts. It might lack the vocabulary and inflections for expressing emotions well. That means it simply doesn't allow C-3PO to say what he wants to say, or at least not efficiently.
Let's take the first line C-3PO speaks to R2-D2 in A New Hope:
Did you hear that? They shut down the main reactor. We will be destroyed for sure. This is madness!
The information that a technical device was shut off and that there is a high risk of non-functionality in the near future can surely be expressed easily in beep-code, because they are useful information for the work of an astromech droid. But the reason C-3PO says this is not because he wants to inform R2-D2 about technical facts. What C-3PO actually wants to express is how this makes him feel. He doesn't communicate this by what he says but by how he says it. The beep-code language might simply not allow this.
When he would try to convey all the information hidden in inflection in the actual text, the beep-code would likely literally translate like this:
Did you hear that? I am certain you did and do not expect you to answer. However, this sound made me really startled and I feel the desire to use it as an occasion to discuss our current situation with you. They shut down the main reactor. I believe that this is a very unwise decision in the current situation. However, I do not believe that they would follow my advise, because they usually do not. The continuous under-appreciation of my technical expertise is frustrating to me. We will be destroyed for sure. I am very afraid of being destroyed. But I do not see a way for me to prevent this from happening. This makes me feel very helpless and desperate. This is madness! I wish all the humans would understand that their violent actions only lead to unnecessary suffering and that friendly cooperation is a much more logical course of action. However, I understand that I will never be able to convince them of this fact. This makes me feel small and inadequate in the great order of things. What is your emotional reaction to the current situation?
Encoding and speaking this whole text in beep-code would likely take longer than speaking it in Galactic Basic. And that under the assumption that it even has words for concepts like "frustration", "fear" or "desperation". Without such words C-3PO would further have to paraphrase these concepts somehow, which would make the text even longer.