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Music is everywhere! It's in our homes, in our cars, in our phones, at work. And to prove it, there is an growing list of applications offering streaming services like Pandora, Spotify, Slacker Radio, Grooveshark, Play Music, Rdio, Last.FM, Rhapsody, and countless other smaller services. Each has their own pros and cons. Due to the major competition in this space, most companies offer free services that are commercial supported, which is fine by me. The two service I use most recently for free streaming radio are Pandora and Spotify.
Web or Software
The most noticeable difference is Pandora's free service only requires your PC browser (running Flash) vs Spotify's free service requires installation of software on your PC. There are advantages and disadvantages the vary depending on your situation. For example, if your office block installing software, your only choice may be Pandora (unless they blocked web access to it). But if you can freely install software on your computer, you can get a richer experience with Spotify. Since I can have access to a richer and more responsive interface, my personal preference is Spotify.
Mobile Apps
Both Pandora and Spotify offer free mobile apps that offer free streaming radio. The Pandora mobile app closely matches their web browser experience. Spotify's offers less features than is found on their downloaded PC software, but is on par with what Pandora offers. Both apps are high quality and as of May 2014 have constant software updates.
Registration
This one is also straightforward. Pandora does not require a user account to listen to streaming radio. Point your browser to Pandora and you can (nearly instantly) start streaming music. On the other hand, Spotify requires user registration (even before you download the Spotify software). When Spotify first rolled out, accounts were by invitation only. Registration then transitioned requiring a Facebook account. Today, it is freely available to anyone with an email address. Therefore, there is no reason not to give Pandora a try. In fact, I suggest you try Pandora's service first.
Software Updates
Since Pandora is a browser based streaming service, you never need to update your Pandora software. The only thing you might need to keep an eye on is keeping your Flash version up-to-date. On the other hand, from time to time, Spotify does update their software. The software is seamlessly downloaded and installed while the application is running. The only time it asks you restart Spotify (not your whole computer) is the next time Spotify launches. So your listening experience on Spotify is never interrupted.
Music Curation
I'm very happy with the music selection from both Pandora and Spotify, but the general consensus is that Pandora is still better at identifying new music that suits your taste. Spotify can make some guesses, but their strategy points much closer to pre-curated playlists of music that are often times hit or miss. Where Spotify excels is providing you the exact song you want to hear. Their strategies vary, but my opinion is the Pandora is much better for discovering new music that fits your taste while Spotify is good for listening to a set list of songs.
Other Free Services
This is where Spotify has an advantage. In addition to radio, Spotify offers you the ability on computer installed version (not Spotify app) to pick and listen to any music you want at any time. It is still advertisement supported, but it gives you the ability to listen to the exact song you want at the moment.
Concluding (My Opinion)
Overall, my favorite streaming radio provider is Spotify. My second most used software is Pandora, but over the years have also tried (but discontinued use on) Slacker, Grooveshark, Play Music, and Rdio. I use Spotify at home and work, and where possible use their app when I'm driving around town.