Il existe- telling someone that something is real. e.g. There are many kinds of dogs. (Many types of dogs exist)
Il y a- telling that there is something somewhere. e.g. There is a cat. (A cat is there)
English does not really specify between the two and my examples aren't that good, but I hope that I answered your question. :)
I just thought of something. I thought of an instance that "There are yellow carrots" would be something I might say. Let's say that I am at a farmer's market and my four year old daughter saw some yellow carrots. She might point at some yellow carrots and say, "Look mom, there are some yellow carrots! I did not know there were yellow carrots!" My reply might be; "Yes honey, yellow carrots exist"
Something like that :) Somehow I feel that in this instance might say; "Yellow carrots do exist"
My Spanish friend has pointed out to me how the English language seems to love all tenses of 'do'. chuckle
I was intrigued by your argument that 'there exist yellow carrots sounds atrocious' I had to smile! You stated no rules. However, I must agree with you, the way a sentence structure 'sounds' is inherently how a native speaker of a language judges whether or not a sentence structure 'sounds normal' to them.
I think this issue causes more difficulty in learning and teaching another language. What are we to do? :)
I think the best answer to your question is what SeaWolven said- it's the difference between 'such a thing exists' and 'look, that thing is there'.
The former would be written 'Il existe', and the latter 'Il y a'. It seems you provided a translation more fitting of a sentence beginning with 'Il y a'. Hope this helps. :)
This is one of the tricky parts about learning French. There is no difference in sound between jaune and jaunes. When you hear "des" (sounds more like like day) instead of "de" (sounds more like duh) then you know it's plural so you'll have to make the noun and adjectives plural too, even though there may be no difference in sound. I hope that helps a little. If you want to feel a little better about it, think of the poor French speakers who are trying to learn English and have to figure out bed, read, and said, which all rhyme but are spelled differently. French is very regular compared to English, which is full of inconsistent things like this :)