Mostly these two structures mean the same thing, and it doesn't matter whether you use the "to do" or the "doing" (the infinitive or the gerund) form of the verb. I couldn't say this is true for every single verb or phrase, but it should cover the majority.
Of course, as with many things in English these sentences can have different nuances depending on the context.
He started running away from me
is different from
He started running because his doctor said it was good for his heart.
In the first sentence "started running" is the verb, while in the second "started" is the verb and "running" acts as the direct object of the verb.
However, if you substitute "to run" for "running" in both sentences, it doesn't change the meaning in any significant way.
He started to run away from me.
He started to run because his doctor said it was good for his heart.
It's also easy to change the meaning if you simply change the time frame:
He started to run a few minutes ago.
He started to run a few weeks ago.
The first implies he's running for some kind of immediate reason, while the second implies he's running as a regular practice. Again, it doesn't matter if you use "to run" or "running" here.