Personally I think a diversified portfolio should have a bit of both.
The bulk of my wealth accumulation was accomplished using real estate, as the careful use of leverage can really boost the capital growth side of things (also can increase the downside in a falling property market of course!). I wrote a bit about it in the posts below:
Another advantage of property is you can potentially "create" value, for example extending or subdividing the asset. This something you can't do with shares.
Aside from these the big difference between property and shares is time. An index fund doesn't phone up shouting that its toilet needs unblocking at 4:30 in the morning! You can mitigate this somewhat using professional property management, but ultimately they will still be asking you questions about repairs and things every now and then.
REITs are a less involved way of getting into property, though without the two big benefits outlined above.
I'll echo @RouteToRetire's suggestion to check out BiggerPockets, there is loads of good free stuff on there... but much like here, be careful taking financial advice from random folks off the internet!