Chinese words usually have multy meanings, words you offer out have different meanings in different sentences, for e.g. Tian means the sky (space) , or the date (time), or huge (large as the sky); and Ri means ths sun, days. Some exmples here: 天空晴朗,万里无云; 天天都迟到,这位新员工大概要被炒鱿鱼了; 天大的委屈都要忍受; 百姓的日子总是难过; 日头已经偏西了,可是爱人还没有回来.

Jeff was coming to the point. 日 was an old word and only a morphem in current Mandarin. It is usually not used indepently today, that is, only appearing in fixed word or phrases: √生日，×生天； √日后，×天后；√ 三月五日（or colloquially 三月五号），×三月五天. √早到了一天，×早到了一日. ......... Sometimes causal writers, Jia Pingwa for one, use old or older words in mondern writing, making a neither-classical-nor-modern and neither-mandarin-nor-dialectal type of phrasiology, that is a poor style.

If it is to mean "day" then 日 is formal use and 天 is more for conversational Mandarin. In Cantonese, 日 is also used in conversations since it uses more classical terms than is modern Mandarin. Even when trying to use these characters with the same intentional meaning, usage may not be interchangeable. The best example is 日子 = days, time. for 天子 = son of heaven i.e. the emperor!

when talking about "day", they mean exactly the same; however, they have other meanings which are very different and cannot be replaced by each other. for example: 日：means sun and something related sun: 日光， 烈日，日月 天：means the sky and something related to sky: 天气， 天空， 天边