Phase-shift oscillators have less distortion than the Wien bridge oscillator, coupled with good frequency stability. That is my opinion (and no flame wars please ). I have added a pdf with some useful information.
When you are designing an oscillator you need to establish the answers to some basic questions.
Is this fixed frequency or variable?
If variable over what range?
Will the oscillator have swept frequency variation?
Is the output to be referenced to earth, a power rail or floating?
What magnitude of output is required?
What impedance levels are required?
Is there a buffer amplifier of suitable quality available or will one have to be designed as well?
What waveshapes are required?
The answers to these questions will help decide on the best circuit configuration for the job in hand as each has its strengths and weaknesses.
In answer to your specific query the Wein is often chosen over the phase shift for variable oscillators as you only need to vary two matched components to vary frequency. With a phase shift oscillator you need to vary three.
Obtaining three (or even two) matching variable capacitors or resistors is expensive and what makes quality signal generators expensive.