From what I gather, speedruns are timed playthroughs of endurance and "perfection" and longplays are just videos of playthroughs from beginning to end.

Speedruns don't allow room for error because of the timed nature of the video. Longplays allow some margin of error, just as long as you complete the damn thing.

I watched half of the War Gods longplay and read in the description that he had to continue 14 times on the last boss or something.

Am I right in saying longplays are done with the sole idea to complete the game instead of complete in the quickest time possible or something?

That would be correct to a degree. In a Longplay the aim is to complete the game as fully as possible, showing secrets etc, whereas in a speedrun you just do the bare minimum required to complete the game.