I think these two sentences have the same meaning, but which one should I use and why?
is used in colloquial language to express uncertainty, doubt and disagreement, usually when asked for an opinion on something.
For example, I wouldn't be surprised to witness the following dialogue:
(Two girls in a shop)
It is basically used as a filler in spoken language, much like a doubtful "Hm", to give the speaker a bit of time to think about how to phrase her doubts. Used like this, it is almost always followed by a pause and a more detailed explanation of the cause of the uncertainty.
The meaning depends a lot on context and tone of voice as well as facial expressions etc.
means that the speaker actually does not know some fact.
As always when it comes to spoken language, the boundaries are a bit blurry and there are almost certainly regional differences.
In a context where one may say "I don't know" in English without any further elaboration, I think "Weiss ich nicht" would be more common than "Ich weiss nicht." However, other phases like "Keine Ahnung" may well be even more common in practice.
1 Weiß nicht.
2 Weiß ich nicht.
3 Ich weiß nicht.
4 Ich weiß es nicht.
5 Das weiß ich nicht.
6 Das weiß ich wirklich nicht.
7 Das weiß ich leider nicht.
8 Keine Ahnung.
9 Ich habe keine Ahnung.
10 Ich habe keine Informationen.
11 Das ist mir nicht bekannt.
Jede dieser Formeln ist möglich. Ein kleines Kind könnte 1 benützen. Im lässigen Umgangston, unter Gleichgestellten, wird man 2-5 oder 8/9 hören. Wenn wir höflicher sein wollen, wird die Formel vollständiger sein. 10/11 ist mehr offizieller Ton.
means literally I don't know. IMHO it's simply not much used like this. It sounds to me a lot like a foreigner studying German. The common phrase for that would be
You can use this exactly as you'd use I don't know in English.
About the phrase:
It literally means "I don't know that," emphasizing that you don't know this specific fact you were talking/asked about.
It's very common in German to change the word order such that the part that you'd like to emphasize is in the first place.
A phrase that I use very often to express I don't know in German is