-Better against cavalry that charges them in the flank (or cavalry with first charge). They just use shieldwall or stubborn. Empire troops might run away. 
This is the sort of thing that I find most interesting. Sure it is possible that they could FBIGO if they are charged in the flank (or with by something with first charge) but it is not likely and it is certainly less likely than state troops. 
Because of dice probabilities LD6 is just considerably worse (3/36 or 1/6) than Ld7.
It is true that they can (at a cost of 31 points) have stubborn. However in order to use it the monk needs to be in the fighting rank where attacks can be directed against him. As he is WS2 (I think unless the Yeoman Guard Grail Monk also gets the WS3) T3 with a 5+ (or possibly 6+ save) directing 2 attacks at him would likely kill him 
2 attacks (assuming 3+ to hit) is .44 wounds at Str 3 (assuming. 5+ save). It is .74 wounds at Str 4 (assuming AP 1 so a 6+ save) and it is 1.1 wounds at Str 5 (assuming ap2). Obviously there are lots more conditions and ways these attacks could be configured but I think these are enough to show that there is a reasonable chance (at least 50%) of the grail monk dying and the stubborn going puff. Then you are relying on your LD6 leadership.
-Better offensively. They are cheaper, have warband, and horde. More likely to make the charge, can go wider for more attacks than empire troops, and have more CR to win the combat. Since their units are cheaper, they are more likely to have extra units nearby to flank the enemy. 
Firstly compared to empire units with detachments they are not more likely to have extra units nearby to flank the enemy. They are substantially less likely to have extra units nearby to flank the enemy and the units they do have nearby to flank the enemy are likely to be worse at the job of flanking the enemy.  However leaving that aside you also don't get to say they are cheaper and have warband and horde and therefore can go wider for more attacks. Either they don't have horde (or some of warband) or they are not cheaper. For example a Yeoman Guard unit that is 6 wide needs 23 models to maintain horde.  A state troop unit needs 17 models (deployed 6 wide) to maintain all its advantages. Therefore deployed 6 wide the Yeoman Guard unit needs an extra 30 points of models to gain horde (and more if it wants to keep it). It needs 26 models to maintain horde if deployed 7 wide. A state troop unit needs 19 models. Therefore to deploy 7 wide you need an extra 35 points of models. 8 wide costs you 40 points of models and so on (an extra 5 points or one model).
However to maintain their ranks are more important to the Yeoman Guard than they are to the Empire State Troops. If you bring the minimum Yeoman Guard then you lose 1CR and 1Ld compared to just 1CR. Therefore Empire troops csn be brought closer to the bone as it were. They are less susceptible to small arms fire and the like.
Re-rolling charge range (the other aspect of warband) does make it more likely you will make long charges. This does give them opportunities for aggression. This is contrasted with the ability to bring concentration of force.
-Better against dragons. They can get two champions, which effectively means another entire turn before the dragon can start eating ranks. They can also use shieldwall and stubborn. 
Sure they can last an extra turn against a dragon by using their extra champion. However what they can't do is defeat a dragon in combat. Empire State Troops with detachments can defeat a dragon with static combat res. I know people are going to respond that the dragon will just charge the detachments but in that instance the dragon will just charge the flank of the Yeoman Guard and suddenly you are a lot less likely to FBIGO on your LD6. To be truthful neither are very good at dealing with dragons but state troops (because they can win the combat and dice game aside the Yeoman Guard can't) are better.  They are pretty much equal at losing (the Yeoman Guard get an extra turn of champion) but the state troops are a lot better at winning.
If empire state troops were the same price as yeomen guard (instead of two points more expensive if armed with the same spears/LA/shield combo), and if detachments could charge with the parent unit in an attack OR detachments could auto charge the flank if they can see it, you might be able to make an argument that there could be one or two situations where empire state troops were better, despite the massive advantage in special rules that yeomen guard get. 
Detachments can charge with the parent unit and (unless your positioning is really really impossibly bad) will automatically charge into the flank of any unit which charges the parent. Now that disruption is back to being US5 some of the old 3*3 detachments might make a re appearance so they can add +2 CR to combat rather than the 7-8 wide I have been using.
Finally I want to consider the matter of price. How much does horde cost. Well on Yeoman Guard Horde costs 25 points. Warband also costs 25 points.  That is to say that you need 5 extra models in order to make use of the rules. However Yeoman Guard get both so therefore they cost 12.5 points each. Well maybe, maybe not, because access to warband and horde isn't all you get for the 25 points.
If you compare units in the following fashion;
How many attacks?
What armour save?
What strength?
What weapon skill?
Then a state troop block deployed 6 wide with halberds is: 6 attacks WS 3 Str 4 Ld7 SCR (Static Combat Res) 3.  US 17 Save 6+ Cost 102 points.
Yeoman Guard:
6 attacks WS3 Str 3 Ld 6(8) SCR3 US 17 Save 5+ cost 85. Special rules: re-roll charges, veteran
Yeoman Guard can then be upgraded to:
6 (12) attacks WS3 Str4 (3) Ld6 (9) SCR4 US 22 save 5+ (6+) cost 132 points. Special rules: shieldwall, horde, warband, veteran
State Troops could then be upgraded to 
Attacks 6(11) WS3 Str4 Ld 7 SCR 3(5) US 17(22) save 6+. Cost 132 points. special rule: detachments.
This is not the extent of this but (aside from the re-rolling of charges and veteran) if you want to use the other rules Yeoman Guard are not cheaper than state troops.