Updated information to what used to be in the FAQ:

What you CAN do as a froob
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* Gain a maximum level of 200. This may take weeks, even months, to achieve, so you'll be playing for a long time.
* Explore almost the entirety of Rubi-Ka.
* Start off at the ICC Shuttleport Island, AKA Newbie Island. There are several items that you can ONLY get there.
* Froobs can participate in Battlestations, which came with the Lost Eden expansion.
* Froobs can also buy Paid Points and use them to receive Booster Pack items such as the Phasefront Hoverboards and Jetbikes. Buying Paid Points does not remove your "Froob" status, and you can continue to play for free, but you should be careful to remove your payment options, otherwise you will get charged for your free account.
* Froobs can now cast and receive Composite Expertise and Attribute nanos, which used to require Shadowlands
* Froobs can trigger Alien Invasion city raids by filling up the City Controller to 100% using CRU.

These are part of the Notum Wars "booster":
* Build towers (control towers, turrets, guardian conductor towers and service towers)
* Participate in tower wars
* Access the zone control buttons on the area map.
* Hear and use the player voices. (as part of the large client)
* Benefit from org contracts (do not confuse with city benefits)
* Access several new vehicle options (Yalmaha, custom Yalmahas, jetpack)

These features came with Alien Invasion expansion:
* Froobs have always been able to use the outdoors facilities in player cities (sky bar, restaurants, whompas to various clubs).
* Froobs can now enter all player city buildings (HQ, shops, Mining facilities, Sauna, etc...) and can use the bank terminal and surgery clinic in the shop and the grid terminal in the HQ.
* Froobs can receive the XP buff from entering the Sauna.
* Froobs can now wear AI social clothes, but cannot purchase them from the Phasefront store or Maria's fashion.  You will need a paid player to buy them for you.

Also:
* Froobs can load Weird-Looking nano crystals and as a result cast improved versions of some nanos, such as Improved Cut-Red Tape.  The nano crystal can only be obtained from Shadowlands.
* Froobs can load several of the nanos that need Victory Points to purchase, most notable Nanite Enhanced Plunder and Divest nanos
* Froobs can receive keeper auras, Shadowbreed effects, Insurance Agent, and Umbral Wrangles.
* Froobs can use the following items which must be imported from Shadowlands:
     -The Original Electronicum in QL up to 50 (QL 1-10 available from backyard vendor)
     -Kinetic Axe in QL up to 25 (QL 1-10 available from backyard vendor)
     -Guard of the Gentleman
     -Jobe Clusters to make Jobe Implants.  (Sometimes the clusters can be found in mission chests)
     -Turn Spirit Weapons (tradeskilled from Inamorata or Sacrosanct weapons plus the appropriate turn spirit)
     -Spirit-Infused Yuttos Modified NCU (tradeskilled from regular Yuttos Modified NCU and Dormant Ancient Circuit)
     -Shadowlands Rings which give signifcant buffs with no requirements
     -Focus-Funneling Spirit and  Primitive Focus-Funneling Helper, (combined with Focus Funneling Devices from Crypt of Home)
     -The Sacrificed Bracelet of Kay
     -Penumbra Tuaq
     -Thar's Armor

What you CAN'T do as a froob
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- Froobs can not make a forum account for the official forums anymore. They were allowed to in September 2009, but about a year later they retracted this right. If you made a forum account during that time you can still post on the official forums.

These options require Shadowlands:
* The Keeper and Shade classes are only available to Shadowlands subscribers.
* Perks are an option that can only be used by Shadowlands subscribers.
* Gain Shadowknowledge and earn levels 201 to 220.
* Access the areas of Shadowlands (obviously).

These options require Alien Invasion:
* Gain alien XP and Alien Experience levels.
* The Global Market search terminals are for AI subscribers only, even though froobs can open the search window.
* Create buildings in player-created cities.
* Use the AI related devices inside, such as player shops, tradeskill machine, orbital strike console, plastic surgery clinic, etc...
* Board alien ships after successfully repelling an Alien Raid. Note that you can still participate in the ground portion of an Alien Raid, but froobs do not count toward teammates getting a ship invite.  Four players with AI expansion in a city raid are required to get an invite.
* Use Alien Invasion yalms.
* Start Alien Invasion city raids using the City Controller to lower the City Cloak.
* Go to any of the Alien Sectors (Sector 7, Sector 10, Sector 35, Sector 48).

These options require Lost Eden:
* Run LE Missions, although you can team with subscribers so that they can run LE missions of their own.
* Use Victory Points, through LE missions and Battlestations. Note that Froobs can participate in Battlestations, and if you subscribe at a later date, the Victory Points that you earn will carry over.
* Access and learn Research. Research is an alternate form of "leveling" that gives a passive bonus to your abilities and skills. Research also occurs in a global scale, as each faction has a number of global research goals.
* Access certain Vehicles. The Battlemechs/Turrets are for LE subscribers only.
* Purchase from the Lost Eden terminals in the tower shop.
* Use the perks and perk actions associated with LE.

Also:
* Froobs do not gain Veteran Points, which are specifically reserved for SLers.
* There are many powerful items that can ONLY be used by those who purchase the expansions. Also, there are a few items that can only be created through tradeskills if you have the expansions. As a froob, be very careful when trading to ensure that you can actually use the item you are about to buy. The same goes for item crafting... before spending cash for the materials, be sure that you can actually craft the item in question.
* The expansions give several major boosts to skills, above and beyond standard leveling and IP distribution. This includes Perks, Research, and City bonus. It is not uncommon for a player with all of the expansions to have a Level 50 character that hits as hard as a Level 80 character in combat, for example.
* As a froob, your rate of growth and overall power will be much lower than someone who has purchased the expansions. XP gain is a lot faster for those who purchase the expansions. This is not because of an inherent bonus on the part of owning expansions, but simply because the monsters in Shadowlands give far more experience than an equivalent area in Rubi-Ka. Also, SL offers unique content at the higher levels (Level 150 or higher), giving higher XP.
* Froobs, as of this writing cannot receive Neuronal Stimulator and Composite Nanoskill Infuses and Mocham's.

I keep looking for this information and I find myself searching for it all the time, and a lot of it is out of date, due to the many recent changes in AO.  I've compiled and updated this information from ao-universe article, Hansoo's FAQ, and posts here the AOFroobs forums.  I'd really appreciate a sticky, and comments with changes or additions to my list.