A ballistic weapon follows a somewhat parabolic trajectory. It goes way up, often into outer space, then it falls down with enormous speed. A bunch of potential energy is built up in its boost phase, and released as kinetic energy in a much shorter dive, usually exceeding several times the speed of sound.

A hypersonic weapon can travel at low altitudes in sustained flight in a non-parabolic flight path. That is the key differentiator. How you get there is up to you.

To that end, that often implies other things. Lower altitudes often has stuff you have to go around: the earth itself (sea level), hills or mountains, buildings, observation posts, radars, or you might want it to take a particular approach path in general for geopolitical reasons and fly it in commercial airline flight paths or along borders. This often means a lot of power is required to sustain flight, so usually not carrying your oxidizer is handy.