I know Mathematica is free for Raspberry Pi, but is some "more Mathematica like" web interface than Wolfram/Alpha out there?

Yes, there is. Take a look at the Programming Cloud. It can be used with a free account, with some limitations. I haven't looked at it in detail as I do have access to desktop Mathematica, so I'm not sure how severe the limitations are, but on a first glance it appears to be a pretty complete Mathematica-in-the-cloud.

I don't fully understand how it differs from Mathematica Online (which has no free version so I never tried it). Here's a thread on that: http://community.wolfram.com/groups/-/m/t/344601

Well, Wolfram|Alpha is not Mathematica and in general, Mathematica syntax does not work with it. W|A has a natural language interface, which makes it possible to ask questions without learning a programming language. But it can also be very frustrating to get it to do what you want (at least I find it very frustrating and unpredictable).

Try telling it to

select primes from {1, 2, 6, 15, 37, 90, 218, 527, 1273, 3074}

Based on the interpretation of this natural language query above, I tried

Select[{1, 2, 6, 15, 37, 90, 218, 527, 1273, 3074}, PrimeQ[#1]&]

which also works.

In[1]:= Select[{1, 2, 6, 15, 37, 90, 218, 527, 1273, 3074}, PrimeQ]

Out[1]= {2, 37}