What is the difference between a Hilbert space and Euclidean space?

According to Wikipedia,

Hilbert space [...] extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions

However, the article on Euclidean space states already refers to

the n-dimensional Euclidean space.

This would imply that Hilbert space and Euclidean space are synonymous, which seems silly.

What exactly is the difference between Hilbert space and Euclidean space? What would be an example of a non-Euclidean Hilbert space?

A Hilbert space essentially is also a generalization of Euclidean spaces with infinite dimension.

Note: this answer is just to give an intuitive idea of this generalization, and to consider infinite-dimensional spaces with a scalar product that they are complete with respect to metric induced by the norm. Clearly, there are finite-dimensional Hilbert spaces, as $\mathbb{R}^n$, with the standard scalar product and Euclidean metric.

Hilbert space: a vector space together with an inner product, which is a Banach space with respect to the norm induced by the inner product

Euclidean space: a subset of $\mathbb R^n$ for some whole number $n$

A non-euclidean Hilbert space: $\ell_2(\mathbb R)$, the space of square summable real sequences, with the inner product $((x_n),(y_n)) = \sum_{n=1}^{\infty}x_n y_n$