A girly romance is a romance where the man, or woman, is willing to give up everything in order to be with the one they love.
A manly romance is a romance where the man, or woman, is forced to give up, or risk, their happiness with the one they love in order to achieve a greater good.
Girly romances don’t necessarily have to be bad (See: “Wall-E”), and manly romances don’t necessarily have to be good (I can’t think of any bad examples off the top of my head, but I’m not a fan of the romance genre generally). But I think the distinction is a pretty good one.
To see the quintessential examples of both, “Titanic” is THE girly romance, and “Casablanca” is THE manly romance.