The discussion on motivation and arbitrariness has shown why grammar is useful in language. Discourse shaped by grammar is more structured than discourse without grammar. Grammar increases the regularity in speech and thus facilitates its processing. It makes the task of the speaker easier, because for him, rules of grammar function like a kind of lane marking: they are guidelines which the production can follow automatically, and thus it is relieved of much effort invested in the composition of the message. Likewise, grammar makes the hearer's task easier because it increases the expectability of what is being said. This reduces the hearer's attention load and thus helps his understanding. In general, the redundancy introduced into speech by grammar reduces the performance load.
If there were no grammar, the significative system of language would reduce to an inventory, viz. the lexicon. All access to linguistic signs would then be holistic. With its rules, grammar allows an analytic access to linguistic signs. It thus supports greatly the composition of messages which do not reduce to the serialization of ready-made pieces, but express complex thoughts. Grammar thus not only makes linguistic communication easier; it also supports creativity.
If there were a universal grammar for all human languages, it could be given a priori and would only have to be followed by speakers of any language. However, language is an activity; none of its components is intrinsically static. The system of a language is not pre-given; it is a by-product of creative linguistic activity:
the question ‘why is there a system?’ can only be answered by saying that the system exists because it is made. ... And this latter ... implies ... that the activity by which the language is made is itself systematic. ... the development of a language is not a constant “changing” ..., but a constant systematization. And every “language stage” represents a systematic structure exactly because it is a moment of the systematization.
(Coseriu 1958; 271f)
The activity takes place in the historical (including geographical, social, cultural etc.) confines of a speech community. The community has its tradition of how cognition and communication by language is done. What already has been systematized may be used by everybody. Therefore, it is also used up, in more than one sense: On the one hand, it loses in expressivity as it is phonologically and semantically reduced, even to the point of zero; so it no longer fulfills a useful function. On the other hand, it becomes old-fashioned; so even if it would still fulfill its function, speakers want to be more creative and do it differently than their fellows did it yesterday. Expressivity requires new systematization.
At the same time, this answers the question why the grammars of languages differ so much from each other: Linguistic activity takes place under the historical conditions of its speech community. Grammars are historical conventions. Even if grammaticalization is partly goal-oriented, there is still a wide choice of possible goals. Grammaticalization creates grammatical structure which is no longer motivated semantically, but partly arbitrary. Partly arbitrary rules form different grammatical systems.