I was giving a prompt containing a link to https://gemini.google.com/ and it successfully gave me the result. (Worked with both Gemini and Gemini Advanced).
But when I do the same via https://aistudio.google.com/app/prompts/new_chat, I get I am sorry, I do not have access to the internet to retrieve content from the given URL. Therefore, I cannot read the article and summarize its key points.
FYI - I selected Gemini 1.5 Pro model from right side bar in Google AI Studio when trying the latter.
What gives and why this difference?
They are different products. The Gemini API gives you access to the language models directly. The Gemini web app is a consumer app built on top of the models that includes other features, including access to other Google products like search and workspace.
If you want to build a connected app using the API, you can supply any tools you want. See the function calling guide and the example cookbook recipe.