Superpositions and probabilistic mixtures are two different ways to combine |0> and |1>.
The superposition a|0> + b|1> describes a linear combination of the two solutions |0> and |1> to some differential equation, but there is no notion of "mixture". A good analogy to understand superpositions is the solutions to simple harmonic motion (like mass attached to a spring). If you start the system from stretched state and zero velocity, it will oscillate like cos(ωt). If instead you give it a "kick" at zero displacement, it will oscillate like sin(ωt). Since any other combination is possible (e.g. kick while stretching), the most general solution to the equation of motion of the mass spring system is acos(ωt) + bsin(ωt), where a and b are the amplitudes. In practice we usually write acos(ωt) + bsin(ωt), so there is nothing "fancy" going on for superpositions: they are just linear combinations of the possible states. (You don't hear anyone talking about a mass-spring system oscillating like cos and like sin at the same time, do you?)
Off topic note for completeness: in physics class we rewrite acos(ωt) + bsin(ωt) as A*cos(ωt-φ), but it's the so you don't see the cos and sin separately, but they are there.
As for "mixtures" those represent our state of knowledge, or rather ignorance. The mixture of 50% |0> and 50% |1> is represented as a density matrix, which has the same "statistics" under measurement. It's hard to do matrices in plain text so I'll cut if off here... but you can look it up.
An even simpler analogy: The difference between a mixture and a superposition is the difference between going either in the North direction or in the West direction with 50/50 probability and going in the Northwest direction.
The superposition a|0> + b|1> describes a linear combination of the two solutions |0> and |1> to some differential equation, but there is no notion of "mixture". A good analogy to understand superpositions is the solutions to simple harmonic motion (like mass attached to a spring). If you start the system from stretched state and zero velocity, it will oscillate like cos(ωt). If instead you give it a "kick" at zero displacement, it will oscillate like sin(ωt). Since any other combination is possible (e.g. kick while stretching), the most general solution to the equation of motion of the mass spring system is acos(ωt) + bsin(ωt), where a and b are the amplitudes. In practice we usually write acos(ωt) + bsin(ωt), so there is nothing "fancy" going on for superpositions: they are just linear combinations of the possible states. (You don't hear anyone talking about a mass-spring system oscillating like cos and like sin at the same time, do you?)
Off topic note for completeness: in physics class we rewrite acos(ωt) + bsin(ωt) as A*cos(ωt-φ), but it's the so you don't see the cos and sin separately, but they are there.
As for "mixtures" those represent our state of knowledge, or rather ignorance. The mixture of 50% |0> and 50% |1> is represented as a density matrix, which has the same "statistics" under measurement. It's hard to do matrices in plain text so I'll cut if off here... but you can look it up.