I',m sorry that Spotify doesn't work when surfing. What's your speed of internet?
I used for ages mobile network and then after good working years at peak time I had this problem. Spotify not working when surfing. I did a speedtest and it was bad. I changed to cable modem 100M/5M (stable connection) and after that my Spotify working every minute no matter how hard I use my internet.
It's suggested to have at least 0.5 Mbps / 0.5 Mbps internet for Premium 320 kbps, but 1M/1M is better or even faster. Upload is very imrpotant for Spotify because it use P2P to share music with others. This need very good Upload to work properly. If Upload is full of traffic, Download will be interrupted (Spotify playback lags). You can try Spotify Web Player, this does not use P2P.
The problem is that it doesnt matter if I am listening to spotify or not when im surfing. I can surf if I just STARTED spotify and didnt start any songs, but if I start listening on spotify, i cant surf. If i close spotify, after 30 minutes I can surf again. Or if I restart my computer.
My maximum speed is 1mb/s download, and 100kbps/s upload.
1 Mbps speed is 128 kbps per second. It is enough for Spotify, also works with some buffering on YouTube 360p. When surfing at the same time, the speed is very low, that's why you get some interruptions. Also Upload is 0.10M, I don't know who provides this kind of upload, at least in Finland even in 1M mobile network upload is 0.5M, same in the 1M ADSL.
Can you try web spotify? If this works better https://play.spotify.com/ , also if possible try to stream songs over 3G mobile network, then you can see where's problem.
I think your speed is low for your Internet usage need. The problem is more in my opinion in your low upload. Is there any chance to get it up?