No MawB, Sweet woodruff is no relation to Woodrush. The Woodrush which plagues our lawns (well, mine anyway!) is Luzula campestris, also known as Good Friday grass because of its flowering time.
However, it is not a grass, which is how you will be able to distinguish it. While superficially resembling a grass, you will see that its growth form is different. The leaves of Woodrush are arranged in 3 directions (not 2 like grasses) so that the plant as a whole, with multiple whorls of 3 leaves, looks like a many-pointed star. Also if you look along the edges of the leaf particularly towards the base, you will see that they have longish cottonwool-like hairs. Most lawn grasses do not have such hairy leaves.