The meaning is the same. 'Start' and 'begin' can be followed by either the gerund or the infinitive, with no difference in meaning.

Su. Ki. is correct about your second example. "Start" and "begin" can be followed by either the gerund or the infinitive without a difference in meaning. However, your first example, with the verb "to stop" is different. "He stopped smoking" means that he used to smoke, but he does not smoke anymore. But "He stopped to smoke" means that he was doing something else and he stopped doing that in order to smoke" For example, "He was walking down the road and he stopped to smoke." This means he stopped walking in order to smoke.