My university has been heavily, heavily pushing the idea of "Webtop" software, in which the computer that the end user uses consists only of a basic operating system and a web browser. All other functions are performed by surfing to a website that has a web-application appropriate for the task at hand. If you wanted to, say, produce a spreadsheet of today's sales, you would go to www.examplespreadsheetprogram.com (or whatever the URL is), and use it from there.
There are a number of advantages to this, most notably the removal of the need to maintain and upgrade anything but the web browser (which is simple enough), and the automatic upgrade to the best version, which "examplespreadsheetprogram.com" will do for you. However, I immediately saw one large, glaring disadvantage to this. And a few others later, but my university has continued to reject all criticism of their beloved plan. "It's the wave of the future!"
While the Internet itself is built with impressively redundant connections and would survive the destruction of 70% of its nodes, connections to 1 particular site, like your company, tend to be more fragile. My own connection to the internet revolves around a shoddily buried cable that goes to my ISP. If an animal were to chew through this cable, I would lose internet access until it was repaired. If I accidentally cut this cable while trying to, say, mow the lawn, no more internet for me. If my ISP were to lose power, or get bandwidth jammed in a DDOS attack, I would also lose access. And with the webtop scheme, the loss of internet access means my terminal is totally useless. Not only can I not send email, read webpages about my job, and so on, but I can't put together spreadsheets, type text, or compose my email either. (Many email clients allow you to queue a message to be send later, a helpful thing to do if internet access goes down. You can write and queue it now and send it when access is restored.)
Now perhaps a company relying on webtop software would do the smart thing and have several ISPs, routed through a local machine that can easily switch between them if one should fail. Even so, there are other disadvantages to webtop setups.
Another one is inability to control upgrades. Many computer users deliberately like to use older versions of software, because newer versions changed the features in a way they dislike. Maybe the new version is more confusing, or it interprets input in a way they find more confusing, or it handles formulas in ways the users no longer understand. If you use webtop software, you use the newest version all the time. If they changed formulas on you and you don't like it, too bad. The old version is just gone forever.
The last issue is storage and control. When I make a spreadsheet at "examplespreadsheet.com," where is it stored? Do I save it to my terminal's hard drive, or does "examplespreadsheet.com" store it? If they store it, and my boss gets tired of paying "examplespreadsheet.com"'s fees, does that mean it's lost to us forever? Worse, what if they then offer it to a competitor to spite us? (Although this may be against the law, I can imagine a more corrupt company doing this.)
I can see why companies would chose to use webtop software, but if they ask me for my opinion, I would argue it is a bad idea. Webtop is probably best for very small companies that cannot afford even a part-time IT department, and don't have any established habits yet. When you get larger, it's probably best to have your own expertise and maintenance on staff. When accounting's complex formula abruptly fails, it pays to have someone who can immediately identify why and start on a fix.