what is the difference between virtual machine classic and virtual machine in azure?

Short answer to your question is Normal VM or Virtual Machines is the new way of deploying your Virtual Machines whereas Classic VM or Virtual Machines (Classic) is the old way of deploying them. Azure is pushing towards the new way of deploying resources so the recommendation would be to use it instead of old way. However please keep in mind that there're some features which are available in the old way that have not been ported on to the new way so you just have to compare the features offered and only if something that you need is not available in new way, you use the old way.

Now comes the long answer :)

Essentially there's a REST API using which you interact with Azure Infrastructure.

When Azure started out, this API was called Service Management API (SMAPI) which served its purpose quite well at that time (and to some extent today). However as Azure grew, so does the requirements of users and that's where SMAPI was found limiting. A good example is access control. In SMAPI, there was access control but it was more like all-or-none kind of access control. It lacked the granularity asked by users.

Instead of patching SMAPI to meet user's requirement, Azure team decided to rewrite the entire API which was much simpler, more robust and feature rich. This API is called Azure Resource Manager API (ARM). ARM has many features that are not there in SMAPI (my personal favorite is Role-based access control - RBAC).

If you have noticed that there are two Azure portals today - https://manage.windowsazure.com (old) and https://portal.azure.com (new). Old portal supports SMAPI whereas new portal supports ARM. In order to surface resources created via old portal into new portal (so that you can have a unified experience), Azure team ended up creating a resource provider for old stuff and their names will always end with (Classic) so you will see Virtual Machines (Classic), Storage Accounts (Classic) etc. So the resources you create in old portal can be seen in the new portal (provided the new portal supports them) but any resources you create in the new portal using ARM are not shown in the old portal.

The Azure Virtual Machine (classic) is based on the old Azure Service Management Model (ASM). Which revolved around the concept of a cloud service. Everything was contained inside a cloud service, and that was the gateway to the internet. While it is still used (extensively) Azure is now moving over to the Azure Resource Management Model (ARM).

ARM uses the concept of declarative templates to configure an entire solution (rather than individual components) So you can create an entire Sharepoint stack, rather than just a singular machine.

ARM also has a much more logical approach to networking. Instead of having a monolithic VM in an obscure cloud service. You have a VM, that you attach a network card to. You can then put the Network card into a VNet and attach a public IP (if you need one)

Unless you have a compelling reason to use ASM (classic) You should create your solution using ARM. As this is the MS recommendation going forward (todo find a link to that) It also means that you can create templates for your deployments, so you can have a repeatable solution.

On the negative, the old portal manage.windowsazure.com can not manage anything that is deployed using ARM, and there are still parts of ASM that haven't been migrated over to ARM yet. For instance you cannot configure Azure VM backup, since Azure backup is ASM and it can't 'see' ARM VMs

It very largely depends on your circumstances though, what it is you are planning for, the method you are going to deploy with. If you are just looking to stand a machine up to do a single task, it makes very little difference. If you are looking to deploy into an environment that will have some concepts of DevOps going forward, then ARM is the way to go.

The one big differences is for resource management. For that new version is called Azure Resource Manager VM (ARM VM).

ARM VM is better in terms of;

• Classic VM must be tied with Cloud Service, and Cloud Service consumes resource limitation and not-so-flexible network configuration.
• ARM VM is managed under Azure Resource Manager (ARM) which can be organized with/without other Azure services. ARM is like a folder of Azure services, and it gives you more fine-grained resource management.

Classic VM can be migrated to ARM VM version, but you have to afford service downtime. To migrate from classic VM, read the official article: Considerations for Virtual Machines.