Song mode (on Elektrons) is a list of patterns, basically. The different between that and pattern chaining is that you can specify more detail about how the patterns are played back - for example, you can set track mutes per 'row' of the arrangement, specify how much of the pattern is played back (number of steps), and loop sections either infinitely or for a certain number of times. You can also jump to a specified point.
A typical way of using it live is to set up looping sections and switch to a new section when you want. E.g. You can have a "verse" section of 2 patterns looping 3 times followed by a different pattern, which leads into a "chorus" section of 2 patterns looping, which jumps back to the verse loop, etc etc. So it's not necessarily just pressing play and letting it do its thing until the end of the song, you can set yourself up with chained/looping sections.
On the older Elektrons like the MM and MD you couldn't do on-the-fly pattern chaining so it's more essential. It's quite possible to use pattern chaining to play live, it all depends how much structure / control you want. Song mode just gives you more options.