Other than making words easier to pronounce, is there any purpose to having the letters divided like this? Should they or a similar concept be added to a constructed language?
This is somewhat similar to "what's the purpose of odd and even numbers".
Looking at the sounds produced in human languages we can distinguish two different ways of articulation, one where sound waves are produced by the glottis, modulated in the vocal tract through opening of the jaw (open/closed) and position of the tongue (front/back), but otherwise pass through unrestricted.
The other type of sound does not depend on the glottis (whose action is optional), and an airflow is generated coming from the lungs, passing through the vocal tract; but this time, there are constrictions at various places of articulation, which modulate the frequency spectrum of the resulting sounds to produce significant differences. Depending on whether the glottis is active, these sounds can be classified as voiced or unvoiced.
The former are called vowels: they are always voiced, and are defined through the positions of jaw and tongue. The latter are consonants, and are defined through the place of the constriction (glottal, dental, labial, ...), the nature of the constriction (fricatives, stops) and whether they are voiced or unvoiced.
As it happens, you kind of need both kinds of sounds in a language to make it pronounceable. Just like you need odd and even numbers to do maths.
You can see this in syllables: a syllable has to have a vowel as its central element, and is surrounded by consonants. You can have different syllable structures: CV, VC, CVC, etc., and depending on language some structures are more common than others. But this is more of an emergent feature than some deliberate decision.
The purpose of vowels and consonants it to make up syllables. We just call the most prominent part of the syllable "vowel" and the the other sounds grouped around that core "consonants". Some sounds can be on both sides: There are languages where the liquids (l, m, n, r) can act as vowels, and some short vowels (i, u) can act as consonants usually named "glides" and given separate IPA symbols /j, w/.
For spoken languages produced the usual way, having vowels and consonants is a natural thing. But there are even natural languages without vowels and consonants: The sign languages around the world don't have an equivalent to this distinction.