I think you are asking the difference between Él continua caminando and él continúa a caminar. The latter is certainly more commonly heard. Most of the time people tend to translate él continua a caminar as he continued to walk, but he continued walking is also acceptable. Él continua caminando is, however, the most direct translation of He continued walking. There is no meaning difference between the two, but there are some stylistic reasons to choose one or another. The real issue here in terms of Spanish grammar is to see the present participle used in a way other than the progressive tenses. In English the second most common use of the present participle is as a gerund or noun form. In Spanish that function is served by the infinitive. These other uses tend to be less obvious and need to be practiced.
Continue is not a verb that is normally put in a continuous tense. This means that the circumstances that would cause you to put it into the continuous tense would probably also be in the continuous/progressive. Spanish uses this form much less than English for continuous verbs, but for verbs which don't normally use the progressive in English the rules are pretty much the same