#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!"