Does the Azure Websites "Always On" option have any implication on price?
Do I pay less if "Always On" is off and page is asleep as wrote here?
I don't think so because you always paying for resources even if you don't use them, but please clarify.
No, it has no affect on the price. For basic and standard modes you're paying for the entire VM anyways.
Keeping Always On helps conserve machine resources since sites that don't get much traffic (*1) get torn down, freeing up resources for other sites to use (*2). However, if you have a site that you want to stay up even during times of no traffic then feel free to enable Always On.
Edit: The only possible way Always On could increase your bill is if you have Auto Scale enabled based on Memory Consumption or something. Having a whole bunch of sites set to always be on might trigger the Auto Scale mechanism to add another instance (VM) to your Web Hosting Plan, resulting in a higher bill.
(*1) "Don't get much traffic" == has not had an http request for the past 20 minutes.
(*2) The first time a torn down site gets an http request it'll get started up automatically.