Firstly, I would say, the 3rd example is incorrect altogether. The difference between “the” and “a” is similar to は and が, in that, there’s a particular feeling of emphasis that each word imparts to a sentence. The air is sweet with the smell of spring. This sentence, to me, feels more like your emphasis is on the end of the sentence. You want the listener to imagine the smell of spring. The air is sweet with a smell of spring. This sentence feels more like your emphasis is on the beginning of the sentence. You aren’t trying to make the listener imagine the smell of spring. You want them to know the air smells sweet, like if it were springtime.

The air is sweet with the smell of springtime The air is sweet with a smell of springtime

#3 is incorrect. #1 and #2 are virtually identical. It is hard to describe the difference in nuance between "the" and "a." #1, "The" suggests that springtime has a smell, it is always the same, so it is "the" smell. The speaker is saying "you can smell that smell today." #2, "A" suggests that although the speaker knows the smell of springtime, they were taken by surprise today. Maybe the speaker first noticed a smell without identifying it: "What's that smell? I smell _a_ good smell." (Sniff, sniff) "Ah, it's a smell of springtime." A common idiomatic phrase for this event is "Spring is in the air." "Spring" and "springtime" can be used interchangeably. "Spring" is the usual term in conversational English. "Springtime" is a little older or more formal. I just double-checked a dictionary, and, the situation is that "Springtime" is defined as "the season of spring"--but "spring" is defined as "the season of the year between winter and summer!"