Well Azure have dropped their prices recently to match AWS. Comparing the two, there's no real difference any more. They've done this to capture market share. Last year Azure was a billion $ revenue generator (Azure actually hosts the iCloud as well if i'm not mistaken). They're expecting it to climb to a $20 billion industry over the next few years.
I've been using azure now for the last few months (have a VM and a few other services that i'm testing), and have been solidly impressed. We're moving all our (and our clients) online presence over to them in phases starting this month.
As Ipwn 4 said... you can't compare it with a dedicated machine, the availability is a key point. Let's take a storage drive that I spin up (virtual drive.. takes a few seconds to allocate another drive and I can choose the exact size I need). The data on there is actually stored across 3 different drives in separate locations in the data center (locally redundant). If I check a box, then it is replicated to another data center guaranteed to be at least 400 miles away (globally redundant option). Compare that with a dedicated machine where I need to worry about hard drive failures and have to replicate databases to mitigate that fact, and cloud hosting becomes a no brainer. Elastic computing is another. I can deploy my web services as cloud services. Most of the time I can have it idling along with a few resources assigned, and when it gets busy, I can let it ramp up or assign new resources (billed by the minute) and just pay for the temporary increase in computing resources. After the busy weekend, I can happily wind it down again.
Cloud hosting really is a whole new beast. The latest toy i'm playing with are their Media Services. I can write a snippet of code that uploads a video, then with a few commands, Azure will transcode it to a bunch of different formats and have it available to play in a number of different formats for different devices. Cannot be easier. It allows smooth streaming and plug in ads as well. Super impressed so far.
On the cost side, we've got a dedicated server with Hetzner Germany, but if I add up all out usage, our costs will actually decrease substantially by moving the rest of our services to azure (which i'm doing this month). Dedicated machines may cost less per unit of compute power, but then you actually need a second one for redundancy, so that pushes the costs way up. Not to mention the human resources of managing all of this. On azure, you are paying for the redundancy, but only for exactly what you use. Also, if you end up spending more than $500 per month, you get a discount of up to 32% depending on what services you use. Making it a very attractive proposition.
You can also run it as a hybrid cloud if you really want (package up that VM image in your office and ship it to the MS data centers, or the other way around). That's a huge plus if you want to change the load split between internal and external resources.
No one can tell me that Azure is crap. After playing with it hands on for a few months, I am extremely impressed with this service. It's the pricing is very good for what you are getting. The prices will only continue to drop as well. Cloud services are going to kill the traditional hosting market, and Microsoft have done one hell of a job in making this a compelling platform. (Azure is doubling in size every 6 months at the moment).