It's not in any way obsolete or outdated.
You use the partitive with a) mass nouns, and b) when you are talking about some unspecified or arbitrary quantity of the thing these nouns define.
I believe the closest thing in English that can illustrate this is biblical language like "come, eat of my food" (vs. just "eat my food"). It's not exactly the same, but close enough.
This is a very vague concept, so in many cases these two forms can be used interchangeably, but there's still a distinction between them.
A couple of examples:
Сначала Гуськов заглянул в каморку мельника и нашел там на полке большую эмалированную кружку, две головки чесноку и полпачки соли, на стене висела ножовка.
Here, "two heads of garlic" is used as a measure. It means "he found some garlic, of which there was two heads worth".
One could just as well use the genitive чеснока here, but this emphasizes that he found "two heads worth of garlic" vs. just "two garlic heads".
И под спиной что-то твёрдое, точно маленькие камушки. Это дольки чеснока.
This means "those were garlic cloves". Here, чеснок is used as an attribute and not as a mass noun. The partitive would not work here.
Using the partitive emphasizes the fact that you're not being exact about the quantity, so sentences using the partitive convey a certain "zing". It's somewhat like "I'll have some tea" vs. "I'll have some of that tea of yours".
Diminutive nouns ending in stressed -ок, like чайку, сахарку, коньячку etc. actually accept the partitive much more readily than the genitive.