When you use an API you instantiate objects and call methods in an imperative manner. On the other hand a good DSL should be declarative, representing rules and relationships in your problem domain, not instructions to be executed. Moreover ideally DSL should be readable and modifiable by somebody who is not a programmer (which is not the case with APIs).
Also please keep in mind the distinction between internal and external DSLs.
Internal domain specific language is embedded in a programming language (eg. Ruby). It's easy to implement, but the structure of the DSL is dependent on the parent language it is embedded in.
External domain specific language is a separate language designed with the particular domain in mind. It gives you a greater flexibility when it comes to syntax, but you have to implement the code to interpret it. It's also more secure, as the person editing domain rules doesn't have access to all the power of the parent language.
They are, in fact, the same thing. DSLs are generally implemented via the normal language mechanisms in Ruby, so technically they're all APIs.
However, for people to recognize something as a DSL, it usually ends up adding what look like declarative statements to existing classes. Something like the validators and relationship declarations in ActiveRecord.