Both of those solutions would allow (and, coincidentally, require) YOU to run your email GUI. Bear in mind you'll need to choose the operating system you prefer -- Zimbra would need Linux, Smartermail needs Windows. Both those software packages have a 'free' version, I believe.
As for power - both your options would be able to handle the usage you've described.. so go for price, I guess.
With the budget you've listed, you might be better off just getting managed Zimbra hosting from someone and not have to worry about installation, configuration and management of the infrastructure. There's a list of hosted Zimbra providers at http://www.zimbra.com/partners/zimbra_hosting.html Pricing details seem to be somewhat scarce, but several of them list packages that start around $45/month.
As for the website component, a VPS should be plenty for this. Again, going with a managed VPS might be a good idea unless you want to experiment/learn about server administration.
If you're looking for some cheap ones, check fuminet.com, renting small server from there. The only issue so far is packet losses.
I've just got server from LiquidWeb, can't tell you much about them yet, but compared to my previous provider (SoftLayer), their network is better. Support? There are few people who don't know anything (at least couldn't help me) about server(s). But it's not a big issue (ticket resubmit always helps, gets assigned to other guy).
I'd recommend you to take a look at beachcomber.net as well, been using them for 1-2 years. I have left them after Pugmarks acquired them for some reasons. Before accuistion, their support was fantastic, now don't know :-)
I would recommend a VPS from a provider that offers the SmarterTools bundle, which gives you a 50-Domain copy of SmarterMail for free. Just get the cheapest VPS from the most reliable provider you can find if the only thing you're going to be doing is e-mail.
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