The important things to remember about SharePoint Online in sense of customizations, design, development, are that 1) there is no Farm Administrator role for the SP Online customer, only the SharePoint Online Administator, and 2) the things you can do with your SP Online site depend partly on the lisence you pay. So what it means, is that you are not able to install anything on the SharePoint server, nor change any server settings in SharePoint or IIS, and also that the license really does matter. Whereas of course, if you host your own SharePoint server, there are no such limitations. (See more about SharePoint Online licensing)
With SharePoint Designer you can do a lot in both (including harm), but there already are some limitations to this in SP Online. The SP Online adminitrator has the most permissions to SPD, as well as the permission to grant or restrict the use of SPD (fortunately, by default, the admin is the only one who is allowed to use SPD).
With VisualStudio (i.e. development tools and possibilities), you run into the most limitations. But don't let them fool you. There's a lot that can be done developmentwise with SharePoint Online too.
- The Sandbox model. It is a subset of the complete SharePoint develpment toolset. I doesn't allow you to deploy anything in the GAC, and it doesn't allow you to access any services. It can use a basic set of objects within the Site Collection. You can make your own web parts, access lists, handle events etc.
Update (Dec. 8, 2011): As of this autumn, Business Connectivity Services (BCS) are available for developers in SharePoint Online (see more about SPO update and BCS & SPO)
For more information on both and the whole picture, see the TechNet Library article on SharePoint Online development.
So how to fit the mobile view into all this? What should you know and take into consideration when planning the mobile usage of your SharePoint site, whether it be Online or on-premises? There is a very good blog post on this subject by Mike Hacker, I won't go into as much detail here. But a couple thought on that subject here too.
First of all, you might want to evaluate the most probable mobile usage for your site. If it is all about team sites and document management, it is probably enough to enable mobile views in your SharePoint environment (enabled by default in SP Online) and make sure that all of the newly created list get mobile view activated as well (the default ones have this feature enabled by default but new ones don't).
If we are talking about a publishing site, or sites, the question of the mobile view, or mobile site, gets whole new angles. Should we force the complete web site to mobile users too (needs some adjustments on the server, so this is not supported in SP Online)? Or should we then create another css stylesheet for mobile users (which is probably the easiest way to make changes for the mobile users in terms of detecting the device, but then again doesn't give the mobile user a choice)? Or make a completely different set of masterpages? And what about our web parts? Are they mobile compatible?
Make it this way or that, my personal opinion is: give the users a choice. Don't force a mobile view on mobile users, thinking that you're doing them a favor by making browsing easier. Not all mobile users like mobile views. And even more so, there's a whole lot of mobile devices with approx 10" displays which is quite enough for a full scale web experience.
With SharePoint Online - on on-premises with the default mobile view settings enabled - it depends a bit on the device which way it will show the site by deafult. As the user with the device, you can switch them either way by changing the ?Mobile= -parameter in the URL string. The zero is the full view, one is the mobile view. So e.g. http://url/?Mobile=1 would be the forces mobile version for the site, whereas http://url/?Mobile=0 would be the forced full version of the site.