Difference Between Equiripple & Least Squares Design for FIR Digital Filters
While I completely agree with Jason R's answer, I would like to add a few things that I consider important. First of all, it is a misunderstanding to believe that for least squares designs the transition band width is smaller than for equi-ripple designs. The width of the transition band depends on many design parameters but it is independent of the optimality criterion. With both criteria you can define arbitrary transition bands ("don't care regions") where you simply do not specify a desired response. This has to be done explicitly for the Parks-McClellan algorithm, but it can (and should) also be done for least squares designs. The most basic least squares design which is to simply truncate the Fourier series of a (often discontinuous) desired frequency response, is definitely no benchmark for comparing least squares designs with other optimality criteria.

The second important thing to notice is that the least-squares design and the equi-ripple design are two extreme points on a trade-off curve between maximum error and error energy. While the least squares design minimizes the error energy, its maximum error is relatively large, and the opposite holds for the equi-ripple design. So it is desirable - and possible - to mix the two criteria. One way of doing this is to use a constrained least squares criterion which minimizes the error energy while constraining the maximum error to some desired limit. In many cases it is desirable to have an equi-ripple design in the passband of a filter and to minimize the (weighted) least squares error in the stopbands, because then the maximum distortion of the signal is minimized, while the noise power in the stopbands is also minimized. The following example shows such a design. It is a linear-phase FIR bandpass filter of order 100, with a constraint on the maximum error in the passband (in this case resulting in an equi-ripple design in the passband), and with minimal error energy in the stopbands. The top figure shows the magnitude of the frequency response in dB, and the bottom figure shows the magnitude of the frequency response in the passband. Clearly, we have an equi-ripple approximation in the passband, and a least squares approximation in the stopbands.