See that bicubic interpolation results in smoother edges? That's a very general statement … but you can find an overview of image scaling algorithms here.
Bilinear interpolation uses a 2x2 environment of a pixel and then takes the average of these pixels to interpolate the new value. It's not the best algorithm, but rather fast.
Bicubic interpolation uses a 4x4 environment of a pixel, weighing the innermost pixels higher, and then takes the average to interpolate the new value. It's – as far as I'm concerned – the most popular.
Area averaging uses a mapping of source and destination pixels, averaging the source pixels with regards to the fraction of destination pixels that are covered. According to this page, it should produce better results when downsampling.
Spline and sinc interpolation use higher-order polynomials and are therefore harder to compute than bicubic interpolation. I don't think the overall increase in processing time is worth using them.
Lanczos resampling involves a sinc filter as well.
hqx as well as 2xSaI filters are used for pixel-art scaling (e.g. game emulators). I don't think there's a good reason for using them in video.