Distant Spell and Spell Sniper do not work.
The sorcerer's distant spell metamagic says:
When you cast a spell that has a range of 5 feet or greater, you can spend 1 sorcery point to double the range of the spell.
Spell Sniper says:
When you cast a spell that requires you to make an attack roll, the spell’s range is doubled.
Having a range of self means these features no longer work with BB and GFB.
War Caster does not work.
War Caster says:
When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.
Since BB and GFB now have a range of self, they target the caster, not "only that creature" as required by War Caster.
It should be noted that lead rules desiugner Jeremy Crawford has stated on twitter that his intent is that War Caster would still work with booming blade.
Twinned spell does not work.
Twinned Spell metamagic says:
When you cast a spell that targets only one creature and doesn’t have a range of self, you can spend a number of sorcery points equal to the spell’s level to target a second creature in range with the same spell (1 sorcery point if the spell is a cantrip).
Twinned Spell doesn't work with a range of self.
Clearer interaction with a new Bladesinger feature.
In the same errata, the Bladesinger Wizard received a change to its Extra Attack ability:
Starting at 6th level, you can attack twice, instead of once, whenever you take the Attack action on your turn. Moreover, you can cast one of your cantrips in place of one of those attacks.
The removal of the phrase "as part of the action used to cast this spell" makes the interaction between this feature and BB or GFB more clear - the spell is not being cast with the Cast a Spell action, rather it is being cast in place of an attack as part of an attack action.
They probably don't work with most improvised weapons.
The material component was changed as well, from "a weapon", to:
a melee weapon worth at least 1 sp
So if you are using an improvised melee weapon, it must be worth at least 1 sp or you can't use BB or GFB. For example, a rock isn't but a table might be. Of course, your DM may interpret "a melee weapon worth at least 1 sp" as requiring it to have a value as a melee weapon - a table might be worth 10 gp as a table but nothing as a weapon. Your DM may also say that a table used as an improvised weapon is not actually a melee weapon at all.
This also means they do not work with the spell shadow blade.
A definitive answer to the question about standing up from prone.
See here: Does standing up from prone trigger the secondary damage from the Booming Blade cantrip?
Now, BB reads:
If the target willingly moves 5 feet or more before then,
the target takes 1d8 thunder damage, and the spell ends.
It is now abundantly clear that standing up from prone does not trigger the secondary damage.
Counterspell probably works.
With the original wording, there was some question as to whether the melee attack was part of the spell (and thus countered by counterspell) or part of the somatic component (implying that the counterspell would counter any extra damage but the attack would still go through). This has been discussed at length in this Q&A: If Booming Blade or Green Flame Blade are counterspelled, does the attack still go through?
With the new errata, the concensus there seems to be shifting toward counterspell stopping the attack entirely.
For further reading, and seeing some interpretations that differ from those mentioned here; some of the above questions (and others) have been addressed elsewhere: