Lyrical jazz dance is a personal expression to the music. It incorporates elements of technique and traditional jazz but is much more expressive and intuitive. Often you are creating a dramatic pattern or you're creating something that is personal to you in a way.
Good lyrical choreographers such as Lea Michele and Travis Wall, they express the lyrics of the artist in a way that can emote to the audience through the dancers movements which is an interesting portrayal because it can relate to the audience in a way that's more emotional and connects with the audience in that way.
Some other aspects of lyrical dance that are important are the quality of breath and the fluidity of movement and also making sure that the dancer is truly connected to the music in an emotional way. They are representation of the voice of the artist that is signing the song and that is an important thing to think about when you're choreographing lyrical jazz dance.
Also technical interpretation is key especially in a competitive realm. Most lyrical jazz dancers are at the highest level of their technical expertise.
That's just a little information about lyrical jazz dance.