The 70-300L is the lighter of the two (by a bit of a margin), but understand that it's not a light lens by any means.
I don't know whether you shoot crop or FF, but some of the crop teles (EF-S 55-250 STM, EF-M 55-200 STM) are super light, plus they're actually quite decent for the money. Price/performance ratio is very high. If you MUST have an L tele, if quality is the priority I'd go with 100-400L; if weight is the priority I would go with 70-300L. A compromise between the two might be the 70-200 f4 IS. Lose some range, gain some length/size but is lighter than both the 70-300 and 100-400, and a bit sharper too.