I'm not a BR competitor at this time, so you'll have to forgive me not being quite as specific in the description.
Benchrest has a number of different formats... kind of like F-Class has Open and TR, regular and Fullbore. There is what is referred to as 'point blank' benchrest (100-200, and sometimes 300yd), and there is long range benchrest.
There are two main 'flavors' of point blank BR... 'group' and 'score'. In group you have a target with a 'record' bull and a sighter bull. You get something like seven minutes to shoot as many shots as you want at the sighter, and five on the record target. This is what a lot of people think of when they think 'benchrest'. The groups are very, very tiny - so tiny that registered matches have to have a moving target backer to prove that there are actually 5 shots in that bug hole. The other 'flavor' is score benchrest. For this you have one large target paper, with 6 bullseyes on it. Again, one is a 'sighter' bull, and you can shoot it up to your hearts content. The other 5 get one shot each, and your score is something like '50-4x' or similar. The 'X' is incredibly small... about the size of the tip of a writing pen. Both varieties shoot individual targets, and then aggregates... usually 5 targets with 5 shots each on them at each yard line.
The above each have 'classes' such as 'Sporter', 'Light Varmint', 'Heavy Varmint', etc. which basically describe the weight and dimensional limits of the gun. It seems like most people end up building a 'Sporter' or LV gun and use it to compete in as many possible classes as they can. Score BR has 'Hunter' class which imposes some unique restraints on the gun... no more than 6x scope, has to have a magazine cutout, and the cartridge has to be at least 6mm and have a capacity equal or greater than a .30-30 - essentially they are like NASCAR stock cars... they have to meet a certain set of arbitrary restrictions, but they *are* full on Benchrest guns, make no mistake! One of the two main governing bodies of benchrest, IBS, allows what they call 'Varmint for Score', or VFS. In this you start seeing people bringing their Light Varmint (hence the name) 6PPC rifles complete w/ 45x scopes and shooting along side the Hunter BR folks. This is also where you start seeing 'factory' or other classes more oriented towards the people with 'regular' rifles who want to come out and shoot. Depending on the club you go to, you may get a cold shoulder if you show up with a rifle without a benchrest pedigree, or you may be welcomed and can shoot alongside for fun even if your gun doesn't meet all the requirements of the rules - how or if the match director adjusts your entry fee can vary widely.
Long-range BR is a little less 'sub-divided' than point-blank BR... first of all, they shoot for score *and* group at the same time on the same target. You still shoot aggs (5 targets), but I believe the folks in the 'heavy gun' category (>17# or so) shoot 10-shot groups, vs 5 for the 'light gun'.
Differences between F-Class and Benchrest?
Some claim there isn't much, and call F-Class simply 'belly benchrest'
as we shoot off the ground, not off of benches. A top-flight F/Open gun and a top level 600-1k BR gun may be one in the same - there is not a lot of difference between the two. Actually, BR has the more restrictive set of rules in this case. It's not uncommon to see people cross over between the two... if you have a gun that can legally play in both games, why not?
As you may have noticed, the most *record* shots a BR shooter may shoot in one string is 10. Usually its 5. F-Class usually shoots 15-20 record shots here stateside, though I've shot as few as 7-10 in Canada. In F-Class, once you start for record, there is no going back to the sighter bull to find out what the changing wind conditions are doing to your bullet impact. You have to suck it up and keep shooting - and actually try to 'read' the wind' (sometimes I might as well be reading braille or Swahili for all the good it does me!) and predict the effect.
Most forms of BR have sufficient time between relays to allow shooters to clean their guns and even reload their cases - in F-Class you pretty much never have time to do more than drag a brush thru the barrel to knock down the powder fouling, if that. Most people don't bother.