Well, 天 is also the word for "sky", and is quite casual when talking about days. 今天 and 明天 mean "today and tomorrow" respectively. 日子 is much more litteral, for example, you'd use 日子 when you're writing, or making a speech, and you CAN also say 今日 and 明日, but they're much more formal :)

天 名词 1. （天空） sky; heaven: 露天 under the open sky, in the open 2. （一昼夜; 白天）day: 前几天 a few days ago; 3. （一天里的某一段时间） a period of time in a day: 天还早呢 It's still early. 4. （季节） season: 夏天 summer; 5. （天气） weather: 阴天cloudy day; 6. （自然） nature: 人定胜天Man will conquer nature. 7. （迷信者指自然界的主宰者; 造物） God; Heaven: 天知道! God knows! 8. （迷信者指神佛仙人的处所） Heaven: 上西天 go to Heaven; die 形容词 1. （位置在顶部的; 凌空架设的） overhead: 天窗skylight; 2. （天然的; 天生的） natural; innate: 天性 natural instincts; 日子 1.（日期） day; date: 这个日子好不容易盼到了 The day we have been looking forward to has come at long last. 2.（天数） days; time 他还有多少日子？He has how many days left(before death)? 3. （生活） life; livelihood 日子在发霉 Life goes mouldy. (indicates boring)

I thought 日子 was a Japanese kid.

天和日子的区别：天一般是比较正式的时间单位，而日子属于口语化的，一般情况下，天比较具体，而日子多用于模糊的概念！

日子 sometimes 日子=天 example 这些日子……=这些天……(they mean a time longer than a day) sometimes 日子=生活 example 他的日子很艰难（he is in a tough life） sometimes 日 is short for 日子 天 most of time 天 means a day example 我用了五天的时间…… 这时候最好不要用日子，听着很奇怪 sometimes 天 means the sky example 今天 天很蓝。 sometimes 天 means God example 哦，天啊！(oh God!) sometimes 天 means heaven example 他升天了(he went to heaven. he is dead) sometimes 天 means the weather（short for 天气）usually has the "er" sound example 今天 天(tianer)不错。(today it is a good weather) sometimes 天 means nature example 他天生就是……。（he gets .......by nature. ） 天 is also used in words 今天(today) 明天(tomorrow) 后天(the day after tomorrow) 白天(day time) 蓝天(blue sky) suggestion 日子 和 天 they ausually confuse foreigners. so remember the sentences they are used in might be a good solution to slove this problem.