Spotify and Grooveshark both allow users to search for songs to listen to, for free, but do not allow downloads, keeping it legal. Both rely on advertisements to keep them going, as well as charging fees for premium users.
The main difference is that Spotify is a programme that can be downloaded to your PC, whereas Grooveshark is solely online, no download necessary.
With Spotify, you will need to register an account, which is no hassle at all, however with Grooveshark this isn't necessary. It is worth noting though, that the user gets a lot more out of Grooveshark with an account. Accounts are free to set up on both Spotify and Grooveshark. Happy days!
Here is the registration screen for Spotify...

As far as searching for tracks goes, both are on a par, understandably there isn't absolutely any track you can think of. If there's stuff not on there that you feel should be, it's up to you to contribute and upload it there!
The interface of Spotify reminds me a lot of iTunes, this is definitely a good thing for me. I use iTunes a lot, so this Spotify setup is easy on the eye, and I've found, after literally 2 minutes use, that it's very easy to navigate around.
This is not to say that the Grooveshark interface isn't pleasing to the eye, I've been using Grooveshark for a while now and I do find it easy to navigate around. It's also easy on the eye and quite clear to the user what is going on.

This is the Spotify homepage, showing what's hot, what's new, a lot like the iTunes store homepage.

Here is the Grooveshark homepage, simple, search for what you want. It gets to the point, it knows why you're there. You do miss out on the whole "new music" side of things however.
So, I've got my account, got my head round the homepage, now I want to search for a song. I decided to search for "Playtawin" by Gang Starr. Thought this might be a challenge for them as it's not exactly pop music. Lo and behold they both have the song, great news. Sound quality is equal for Spotify and Grooveshark, both more than adequate.

Here is the "Now playing" screen for Spotify. Again, it looks a lot like iTunes, nice large album art in the bottom left corner, not much else to say really.

Here is the equivilant for Grooveshark.
You will notice that I have the info open for the track. This is a really useful feature, it provides the direct link for the song, so you can share it with friends. It also shows similar tracks, which is a good idea. The only problem with it is that in my experience, the similar tracks it throws up aren't similar enough. This will no doubt improve as it grows and more users contribute to Grooveshark. You may notice the play icon that appears to have a jagged circle around it. This is autoplay, and will play endlessly throwing up similar tracks, but again, it's what they feel are similar tracks.

Talking of sharing the music, here are the sharing options for Spotify, Delicious and Facebook. Quite limited, but it is very easy to share what you are listening to with other Spotify users, as it provides a URL you can share which will automatically open up Spotify and go direct to the track/album/playlist that has been shared.

Here are the sharing options for Grooveshark. You have Facebook, Twitter, and StumbleUpon. As a big Twitter user, this gives Grooveshark a big advantage for me. You can tweet what you have listened to, it will automatically shorten the URL via bit.ly and you can add a personal message. Fantastic!

A big disadvantage for Spotify - the adverts! It will play adverts between songs. Not between every song, but it is annoying and you can't skip them. With Grooveshark, the only adverts you see are on the sidebar, and to be honest, you hardly notice them.
It's worth noting that during writing this blog a really loud condom advert came on, very embarrassing.
Spotify does also have a radio, although I found it very difficult to understand, I was expecting it to be the same as last.fm, but it really isn't. I will continue to play around with that and will address that in a future blog post.
So, to conclude, I really like the look of Spotify, but I do find Grooveshark more practical. It's a lot like comparing any leading mobile phone to the iPhone. Grooveshark is the high spec phone that works really well, and has all you need and perhaps more. Spotify is the iPhone, that may not be as good in spec, but it does just look and feel so good. It's also nice to have an application you can open rather than having an internet browser open for it.
I definitely think that Grooveshark's sharing options are a lot better, you can share it with anyone who has an internet connection, and they will not need to register. The twitter sharing really is a plus as I mentioned earlier.
If everyone in the world had Spotify, and contributed to it, it has the potential to be a massive service and fantastic for music sharing. But as it stands, Grooveshark is a lot better for that.
I'm going to make a decision, and because of the advertisements, Grooveshark wins this battle. The Twitter sharing also bumps up its score. It would be a lot closer had Spotify not bombarded me with a condom advert, bad times.
I will end though by saying the interface on Spotify is beautiful, making it a joy to use. I will continue to use it for the forseeable future. I will just have to keep my eyes and ears open at the end of each song to save further embarrassment.