In general, there is not a great deal of difference between these. For example, some dictionary definitions of "law" say that it is a set of rules. There are, however, some differences.
The term "law" generally refers to a rule (for want of a better word) that has been formally made to apply to everyone in a society. In a democratic system, a law is something that has been passed by a legislature. In other words, a law is much more formal than a rule.
There is another, more technical, difference that is used in the United States, at least. A law is something that has been passed by the legislative process. By contrast, a rule is something that has been created by the executive branch. The legislature passes laws that give the executive branch the authority to make rules. The rules are enforced just like laws. However, the legislature can make laws that would abolish various rules while the executive cannot make rules that abolish or change the basic meaning of a law. This definition can be seen in the link below.
Overall, then, there is not much difference. The word "law" generally refers to things that are more formal and carry more weight. However, the words are often used interchangeably.
Rules are judicial legal findings as in a judge's ruling on a case. It is the legal principle or principles established in a legal dispute.
Laws are made by legislatures such as Congress.
Regulations are made by the executive branch usually through the various departments such as departments of state, agriculture, treasury, etc.
Regulations are similar to rules in that they provide more specificity to a particular law passed by a legislative body.
Of course there are other non legal uses of the term "rule".
Rules: are codes of conduct that are designed for specific situations, similar to customs but have a greater importance because there is usually a punishment associated with them (e.g. if the rules of soccer are disobeyed, the individual that breaches these rules may be issued with a red or yellow card)
Laws: are legal rules of conduct created and enforced by a sovereign power such as the state. Laws apply equally to all people- this is known as the ‘rule of law’.