What is the difference between smart antennas and phased arrays?
Simply put: both are different ways of harnessing the properties of using multiple antennas.

In a Phase array the antennas are usually placed in a repetitive pattern like a matrix or circles or hexagons. Often the distance between the antennas is always the same. Phased arrays are used to make these many antennas behave as one antenna with certain properties like a more narrow beam pattern (where the antenna is sensitive and/or emits more power).

I find smart antennas a misnomer as the antennas are not smart at all, the "smart" bit is in the signals which are used with the antennas. This can be used for improving reception by beam-forming or the reverse: enhanced sensitivity in a certain direction.

This "smart antenna" signal processing is also used with Phased Arrays.

You could say that some smart Antenna systems are a simple form of Phased Array. Where a simple Phased array might use 9 antennas, a smart antenna system might use 3 antennas.

You could call any Phased array with signal processing a Smart Antenna system but I would not call any smart Antenna system a phase array. for example some laptops have 2 or 3 Wifi antennas for beamforming and Mimo. You would not call that a phase array but I think it would be OK to call that a smart antenna system.