For English native speakers, can you easily distinguish people who speak English as their mother language from those who don't based on the writing style?

Sometimes yes and sometimes no. It the non-native speaker has a high standard of English then it would be very difficult, if no impossible. Generally you can spot a non-native speaker by the errors they make. While some native speakers do make errors, they will make different kinds of errors. Recognising certain types of errors can give away the writer's cultural background.

For example, a writer who mixes up then/than would be an American with native English skills. Where as, writers who often omit the space after a full stop or put the space before a full stop, show a likely Chinese cultural background. (When typing in Chinese there is no need to insert spaces between sentences as the Chinese 。contains whitespace in the character block.)