Azure CLI is a PowerShell-like-tool available for all platforms. You can use the same commands no matter what platform you use: Windows, Linux or Mac. Both, Azure CLI and the PowerShell package use the REST API of Azure.
There are two version Azure CLI. The Azure CLI 1.0 was written with Node.js to achieve cross-platform capabilities, and the new Azure CLI 2.0 is written in Python to offer better cross-platform capabilities. Both are Open Source and available on Github. However, for now, only certain PowerShell cmdlets support use on Linux.
PowerShell Command: Connect-AzureRmAccount
Azure CLI Command: az login