Private forums are usually forums that only members (either all members - or only a certain group of members (example - maybe you have a private forum for moderators to discuss new features)) can view - public forums are available for anyone to see but you can make it so that only registered members can post to the forum.
Dealing with spam is a constant pain in the *** for mods and admins but you reduce it by making it so that only registered members can post to the forum AND making sure that any one registering has to authenticate their email address - that way they can't just use a mickey mouse email address to become a member and then post spam.
This is not fool-proof however and a large busy forum will still attract spam - people can setup free email accounts and then use that account to register with any forums they want, authenticate it and then spam the forum.
Mac-forums is not spam free - I see spam on most days but the mods or admin or iNet people clean it up pretty fast.
Having a spam free as well as (hopefully) a trouble free forum is all about the moderators - people that can spare time to help the forum along and to remove any offending posts.
You can of course only allow people to sign up after you've ok'd their application in most forums - this is however a slow process and unless you really need to restrict who has access to the content - then it's not really recommended as it will give you more work and drive
potential members away because by the time you've managed to get round to accepting them as a member - they've got bored and found another forum that can provide the answer a little quicker.
I've not used freebb so can't say what it is like - but as it looks like it's based on phpbb then I'd guess you will have the same administrative options as a phpbb forum so you can control a lot to help you deal with potential spam (blocking words, blocking IP addresses etc)
There is one major benefit of having a public forum over a private forum and that is Google - well all search engines in fact - if you make the forum private - then the search engines spiders will not be able to take a look at your content - they can't index what they don't know about - and you will not be able to catch users looking for answers to questions you've already answered...
typing seems fine - baby most be quite good