While DocRoot is correct in that sometimes the only difference is marketing, in general there are some concrete improvements that most hosts make to the WordPress and server environment) that are used for these plans.
On the low end (DreamPress, GoDaddy, 1&1, et al) you will likely be placed on a VPS that has Varnish configured to run somewhat optimally with WordPress. The “managed” aspect usually means that the server is monitored for performance and the host makes sure the core is updated (which is less important that it used to be since it became a default). If a hosted site slows down or begins consuming too many resources, the host usually will notify you and/or take steps to reduce the server load.
The next tier up from these basic plans usually includes some form of security assistance and preventative features and some will also claim to have a better database server (exactly how is hard to quantify). For example, I believe GoDaddy struck a deal with Sucuri not too long ago to integrate Sucuri’s firewall and server checks into the GoDaddy WordPress hosting plans. Dreamhost use to use StopTheHacker but I haven’t seen mention of it recently so I don’t know what went on there. Other extra features would include some form of basic CDN or valet support for the site.
Beyond these basic plans are the Dedicated WordPress Hosts, such as WP Engine, Page.ly or WordPress’s own VIP service. These services are roughly similar to the basic ones but tend to have more features (staging environments, better security, backups, etc.), way better support and a much more thorough approach to optimizing and securing WordPress and will also cost a ton more. WP Engine is famous for curating plugins and themes that have known performance and/or security issues and won’t even let you install those on their site. Whenever you attempt to install or update something, they prompt you to create a snapshot backup for rollback purposes and have live chat and phone support if something goes sideways. That’s the kind of service you get for the upper end tiers.
Are these things any better than a “standard” *nix hosting plan? Depends. If you have shell access to your server and enough permissions to install your own stuff or change how PHP is deployed AND you know what you’re doing with hardening WordPress, then the answer is “probably not” as you have the experience and expertise needed. If you don’t know how to manage the hosting environment to that level or don’t know/understand the security ecosystem around WordPress, then you probably should opt for whichever managed setup gives you the best bang for your buck.
For your particular case, you very well may want to explore the higher-end tiers of managed hosting as speed and security are more critical to e-commerce than other types of sites but it all depends on how much time and energy you want to dedicate to the environment instead of the site itself. If you can do both, great! Save yourself some money and get a good vanilla VPS or dedicated box. If you are better with WordPress than server stuff, go with managed hosting or a dedicated host.