Now since the example is "I" being the subject, so there're no difference of sentence structure using either 自信 (self-assurance, confidence in yourself) or 信心 (confidence in yourself or someone/something else). The difference will occur in a sentence like this:
Traditional: 人們對今年的經濟前景沒多大信心。
Simplified: 人们对今年的经济前景没多大信心。
Pinyin: rén men duì jīn nián de jīng jì qián jǐng méi duō dà xìn xīn。
Meaning: People don't give much hope to the economic growth.
Note: In this sentence, don't replace 信心 with 自信, though I can't say it's utterly wrong to use 自信 here.
In your example, you can use both 信心 and 自信, because "confidence in oneself" is the common point between these two words:
Using 信心
Traditional: 在舞臺上跳舞,我沒信心。
Simplified: 在舞台上跳舞,我没信心。
Pinyin: zài wǔ tái shàng tiào wǔ,wǒ méi xìn xīn。
Using 自信
Traditional: 在舞臺上跳舞,我沒自信。
Simplified: 在舞台上跳舞,我没自信。
Pinyin: zài wǔ tái shàng tiào wǔ,wǒ méi zì xìn。