Negation in mandarin?

The word “沒” is just a short form for沒有, so the meanings are totally the same.
The nuance is on the mood; basically I don’t suggest you to say 沒, it sounds you don’t want to talk to the person, so you use less word to express that you have no interest to say more. Sometimes it even shows apathy or angry.
“不” is a strong mood to say “I don’t agree, I don’t think so, or I don't want, etc.”. Normally, when someone say “不”，he or she will continue to say the reason, why is he or she against it. In your conversation practice, you should avoid this short word by saying “I don’t think so – 我不認為如此/wǒ bù rèn wéi rú cǐ”, “I don’ agree – 我不同意/wǒ bù tóng yì” , "I don't want - 我不要/wǒ bù yào" or “I have an alternative thought – 我有不同的想法/ wǒ yǒu bù tóng de xiǎng fǎ”, etc.