Picking on someone implies merely bullying -- unfairly making fun of or harrassing someone because of something they can't help, such as their physical appearance or mannerisms. An example would be a short person being harrassed because of their physical size, or a person with a mole on their face being harrassed or made fun of because of the disfigurement. Or because of their religion or nationality, etc.
Ripping on someone involves severely criticizing their actions or their political positions, or even their work, such as your boss ripping on you for your bad job performance. Ripping may be fair or unfair, but it is generally directed at actions or attitudes.