Generally speaking: 'like+gerund' is British English and 'like+to+infinitive' is American English. In British English, we use both forms, but there is often a subtle difference. If we are talking about activities which we enjoy, we use 'like + gerund' eg I like swimming. The form 'like + to + infinitive' exists in British English, but it often has a slightly different meaning. Here are some examples: I like to see my dentist every six months. I like to finish my homework before dinnertime. I like to go on holiday in September rather than August. These are not activities which you enjoy as such, but you know that they're good things to do. You probably don't particularly like seeing your dentist, but you know it's good to keep your teeth healthy. You don't enjoy finishing your homework, but it's a good feeling to get it done early so you can relax for the the rest of the evening. The third sentence isn't about how much I enjoy my holidays, but about the fact that I think it's better to take them in September - perhaps it's quieter then ie I feel it's a good thing to do. American English, by contrast, tends to use the 'like + to + infinitive' in all cases. For example, an American would probably say 'I like to swim' as an example of an activity which he/she enjoys. That said, there are some overlaps. The American usage is possibly becoming more common in Britain, as a result of US films, TV and the internet.

There is no difference in meaning, but these constructions are used differently in American and British English. The construction [verb] + [-ing form of verb] is more common in American English. For example, "I like playing football." The construction [verb] + [infinitive form of verb] is more common in British English. For example, "I like to play cricket." However, all English-speaking people around the world will understand both constructions.

Whilst "to + infinitive" is generally recognised as American English and "gerund" is technically British English, both are interchangeable. However, in British English we generally use "gerund" when talking generally and "to + infinitive" when talking specially. "I like like swimming." vs "I like to swim in a lake on Sunday mornings." This is true for other verbs of choice or emotion too (i.e. love/hate/prefer/dislike etc)