They are often used interchangeably to show possession, but what is often missed is that "have got" is a present perfect form. This implies that something happened in the past to explain the current possession. In some cases, by using "have" you suggest it is a natural state. "I have brown hair" is natural. "I've got brown hair" means you probably dyed it. For money and other things you acquire, either form may be used. The "I've a..." form is still used, but I also have misgivings about using it.

Both 'have' and 'have got' are used to show possession in the present. The only difference in this sense is the structure in negatives and questions. With 'Have' we add the auxiliary 'do' like all other verbs whilst with 'have got' we don't. so 'I don't have a dog'. 'I haven't got a dog'. 'Do you have a dog?' 'Have you got a dog? With 'have' there is no contracted form in the present we use 'have got'. ' I've got a dog' not 'I've a dog' 'Have got' is usually only used in the present simple. We use 'have' for the past and future forms. 'Have' is also used for many actions. Have lunch, have a shower, have a holiday etc. 'Have got' is not used in any other sense except possession. 'Have got to' is also used like 'Have to' show obligation.

Firstly, both your sentences are wrong! 1. I have A pen. 2. I have got A pen. There is NO difference in meaning.