Kyotan 你好,
In fact, the two sentences could be understood
very differently, and it's all about their
implications.
我的汉语不好 (literally: My Chinese is not good) could be understood as 我的汉语不太好 (My Chinese is not that good), which has an implication that you meant to be
modest (谦虚) and respectful (尊敬) to your audience.
我的汉语很差 (literally: My Chinese is very bad) is, first of all, a positive statement, and people know you were stating a fact concerning your Chinese proficiency. Moreover, people might assume that you were having a pretty
negative attitude with yourself (e.g. you were sad with your Chinese proficiency).
In my perspective, a typical Chinese who heard "我的汉语很差" would probably negate it (没有啊，你的汉语挺不错的 = Oh I think your Chinese is actually pretty good) or console you (没事，我觉得你学汉语肯定没问题的 = No worries, I think you'll be just fine with your Chinese), in order to make you feel better.
So I'd recommend you 我的汉语不好 or 我的汉语不太好 if you don't meant to express a negative sentiment toward yourself. Or even more explicitly, "
我是汉语初学者 = I'm just a beginner with Chinese", since it sounds more neutral (although not idiomatic) and do not contain any further implication.
This phenomenon of "
委婉/隐含的意思 ＝ implied intentions" could be found in most, if not all, cultures. For example, I think it's really similar to "
裏読み" as in the Japanese culture or "
implications" as in the American culture.
Hope this answer could be helpful to you and good luck with your Chinese!
我的汉语不好/很差 understandable
我汉语不好 better
我汉语说不好 much better
我汉语说不好 could be misleading in some way.
"说不好" is equivalent to "说不准 = not sure" in the North (and maybe the South?), especially in Beijing.
I guess we might want to add a "得" to make it "我汉语说得不好"
