This is the coolest part. Those synapses are the reason the brain is so complex! Basically you've got the first part right, the neurones are quicker and they transmit messages from one end to the other. The other thing you have to do is analyse and calculate. Signals from multiple neurones feed into a single neurone using a chemical synapse. Similarly the converse is true, a single neurone can feed multiple synapses. So when we for example walk, receptors monitoring our balance can feed this to our conscious and unconscious mind, but I'm grossly simplifying this. What our brain does is take hundreds and thousands of ports of information, with where the chemical synapse is and the type of synapse and the receptors and a million other things just slightly changing the information until it's perfect and then send that information a million ways however it likes.
The most important thing is this allows inhibition. Most of the brain uses inhibitory GABA rather than activating signals. Furthermore, the system uses the delay created. Different transmitters act for different durations and send a different volume of signal. An impulse is all or nothing, it's of a fixed amplitude, however a chemical signal can be fine tuned. Neurones can not only interact to inhibit themselves or other neurones to increase the resolution of a signal etc. Conversely a system of just physical connections allows little regulation, in fact it'd be practically a seizure.
Does that make sense?