Well, without considering some some specific cases and actual usage-frequencies for different meanings of these words, only comparing their sets of allowed meanings, it appears that частка has more meanings than частинка.
The generic word частина (not mentioned in your question) has two main meanings, which both are usually translated into English as part (either a physically separated part, or a logical part of something). Additional meanings include part (the largest structural unit) of a book, organization/company division and military division.
The word частинка derives both main meanings of частина — usually meaning small part (because the word частинка is a diminutive from частина). The additional meaning of частинка (comparing to частина) is particle as a minute fragment or quantity of matter.
The word частка has all the meanings that частинка has, and in addition can mean:
- section of a leaf;
- quotient (division result);
- grammatical particle.
So theoretically both частинка and частка can be used to refer a subatomic part, although practical usage-frequencies may differ.