I continue doing it. I start doing it. I continue to do it. I start to do it. В чем разница?

Как Вам уже правильно ответили выше, разница между START TO DO и START DOING something несущественна, и зачастую едва заметна, оба варианта грамматически правильны.
Однако, если покопаться, можно заметить некоторые отличия. Вот что удалось найти 1) Первое отличие - http://forum.wordreference.com/showthread.php?t=1448036&langid=18

The man started to run. --> but he finished doing so

The man started running.--> and probably is still running.

He started to run before he ran, but once he is running, we say "He has started running".

The difference is usually slight, but real:

He has never actually run for public office. The man started to run but did not seem to be able to get all the requisite papers filed in time. [Running does not fit this sentence.]

The man started running back in December, and he still runs every Wednesday night. [Running fits this sentence better than to run.]

2) Ещё один аспект - http:// www. usingenglish. com/forum /ask-teacher /45606 - difference-between-start-do-start-doing. html (убрать пробелы) start (v.)

This weak verb can be combined with either a gerund or to plus an infinitive: He started speaking and He started to speak mean the same.

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Note, though, that for some native speakers there is a difference between start to do and start doing:

Start to do - hasn't started yet or is in preparation

Ex:I'll start to do that tomorrow.

Start doing - has started already or is already prepared and ready to start

Ex: I'll start doing that tomorrow.