Wi-Fi repeaters (range extenders) do indeed improve the coverage of Wi-Fi. What is less well known is that there's a speed penalty when running the range extender in Wi-Fi mode. Catching the router's Wi-Fi signal the rebroadcasting it has a speed penalty. It halves whatever Wi-Fi connection speed your router can provide. This might not matter if all you're doing is watching a bit of BBC iPlayer or ITV X. However, if you have a house full of wirelessly connected devices in regular use then you might want to consider a wired connection.
TL

R - no, a wireless repeater isn't as good as a piece of Ethernet cable. You'll lose half your transmission speed going via an extender.