The simple answer is that a suitably-designed phased array of the same size (same cross-sectional area) as an equivalent dish would be expected to perform approximately equally (in terms of directivity and signal-strength). On those basic measures, it can't really perform better than a dish.

In a receiver, the signals from the individual array elements are electronically summed (at some stage of the signal processing) with subtle time-delays between the elements in order to "phase" the array and create the required directivity. Signals from the wanted direction will sum constructively, while signals from other directions will tend to be non-coherent (at random phases) from the different received elements, and thus average down to a proportionately lower level in the summation.

The key advantage of phased-arrays is that the beam-direction can be "electronically" steered (i.e. by changing timings in the signal-processing), as opposed to having to physically move a dish. The electronic beam-steering can be essentially instantaneous, whereas the rate at which you can move a dish on motors is limited by physical mass, inertia, motor-power, ... and will be subject to mechanical wear. This high-speed steering is near-essential for tracking low-earth satellites, or military radar, among other applications.

There may then be second-order benefits such as physical simplicity, lighter-weight, less wind-loading etc. But if you need super-speed scanning, then moving a physical dish of more than a certain size is simply impractical. Very large dishes are major engineering projects, as the dish needs to retain it's shape to within a fraction of a wavelength (say 1/10th of a wavelength) as the dish is moved and steered. In contrast, a phased array can be fitted on a flat (or even uneven surface/terrain) and "electronically" flattened (corrected for physical distortions).

A further benefit of phased arrays (for relevant applications) is that you can double up (or triple, or...) in your signal-processing, and then receive from two or more directions simultaneously with the same physical antenna. That's something you simply cannot do with a dish. Again useful for Starlink-type applications where you have multiple low-earth satellites, or military radar when you want to track multiple fast-moving targets.

If you have a phased array with a lot of elements then you may be able to control side-lobes better than with a dish, which may be important if you not only want to maximise the signal strength of a wanted signal, but also reject or suppress an unwanted signal of the same frequency but coming from a different physical direction.

Phased array antenna with many elements are likely to be more expensive than dishes, often considerably so, although the cost of RF electronics is continually falling.

(I don't have any first-hand experience with phased-array antenna, but have a lot of experience with signal-processing in other applications, where the underlying reasoning would be similar)