What exactly do all the extra modes on my oven do?
Okay, both of the answers from Greybeard and Manu have part of it.  I know this is an old question, but I assume others with fancy ovens may wonder about such settings too.  Let's put it all together.


ROAST - note that for this oven, ROAST is a convection setting.  Although the manual doesn't state this explicitly, generally convection ovens with a ROAST setting combine heat from the convection element along with the top and bottom heating elements that would normally be activated on a non-convection BAKE setting.  The difference is that convection BAKE usually employs only the heating element near the fan (the convection element) to avoid excessive browning for baked goods and such.  Thus, in convection BAKE, the heat is primarily transferred only via air circulation.  But in convection ROAST there is radiant heat from the top and bottom elements as well, which will help to brown meats and such.  (For the record, BROIL under convection will typically employ the top heating element as well as the convection fan element, but no bottom element.)
EASY CONVECT - this one is explained clearly in the manual.  Or, well, maybe it's not explained clearly, but the explanation there implies what it does.  Essentially, you need to make adjustments when cooking via convection compared to traditional baking.  The air circulation can cause the surface of foods to heat up faster, which may lead to excessive browning (e.g., for many baked goods) or excessive drying out (e.g., for meats in some circumstances).  Normally, you need to convert a recipe from "standard" non-circulating air to "convection" yourself, usually by adjusting the temperature down (often by 25-50 degrees F) and sometimes decreasing cook time.  This oven allows you to simply enter the type of food you have -- MEATS, BAKED GOODS, or OTHER (which includes convenience frozen foods) -- and then enter the typical temperature and time you'd use for a standard recipe with a normal non-convection oven.  In this case, apparently the oven won't listen to you, but instead will do some magical math conversion that decreases the time and temp to try to approximate the way you should cook the dish using convection.  (For example, you tell your oven to bake at 350F for 30 minutes, and it instead chooses to back at 325F for 20 minutes.)  So rather than guessing on how to convert a recipe from "standard bake" to convection, you can let your oven guess for you!  Easy!  (Apparently...)
SLOW COOK - This one is the hardest to find clear info on, because these settings tend to do different things in different ovens.  This review of a Kitchenaid oven (which may be the model OP has) with the SLOW COOK setting claims it "keeps food just above the keep-warm temperature, providing an alternative to a countertop slow cooker."  I believe the clue here is in mentioning the KEEP WARM feature.  You see, most modern ovens have a minimum temperature of about 170F, because lower temperatures will not heat food fast enough to prevent food poisoning (and can allow it to remain in the danger zone below 130F for far too long).  Apparently the KEEP WARM setting on some Kitchenaid ovens however can go as low as 145F.  (Note in the manual that these settings come with a big warning about food poisoning when using them.)  The important thing is that these fancy Kitchenaid ovens are capable of lower heat settings.  Anyhow, one complaint about oven SLOW COOK settings compared to crock pots is that they often heat too quickly, and the food thus spends too much time at a high temperature.  My guess from the description in the above review is that these ovens in SLOW COOK mode attempt to avoid this problem by gradually increasing the temperature over a longer time span.  Another possibility is that I've seen some ovens even disable the possibility of temperatures lower than 200F or even 225F (again, likely to prevent food poisoning).  I can't find any specs on Kitchenaid ovens that state the "normal" temperature range allowed for the BAKE setting, but it's possible that they only allow very low temperatures (as low as 170F) to be entered specifically for the SLOW COOK setting, ensuring that people are consciously employing it and supposedly following the manual's recommendations for time and temperature for safety.  (I had a fancy slow cooker a few years back -- might have even been a Kitchenaid, but I can't recall -- that wouldn't even allow you to set it on "Low" with a time less than 8 hours, likely for similar legal liability reasons.)