Why should cognitive semantics and formal semantics co-exist?

Cognitive and formal semantics can coexist very easily. They not only have different goals but also different objects of study.

Formal semantics grew out of formal logic and is concerned with a very limited set of statements that can be truth conditional (about which the truth can be determined). Formal semantics is not much concerned, for instance, with lexical meaning but looks mostly at combinatorics at primitive units. Formal semantics is the natural choice for people working in rule-based natural language processing.

Cognitive semantics is just a particular perspective on more traditional semantic concerns and tries to root meaning in human experience of the world which it finds in human cognition. It does not deal with primitive units of meaning and therefore is not concerned as much in their combinatorics. However, it looks at how meaning is constructed in speech, so it is not limited to small scale utterances. As such it is more difficult to formalize and use directly in natural language processing.

Both of these approaches to semantics co-exist with more general semantics which looks at both lexical and relational phenomena in language as they are found rather than through a specific lens but borrows from both of these as needed. Modern stochastic-based NLP is more likely to draw on both as appropriate, as well.