I'm not entirely sure what your question is here, but to confirm: The Clone Wars takes place between Epsiodes II and Episodes III. Lucas has confirmed this himself in interviews:
Q: Why is it important for you to tell the story of The Clone Wars?
A: The interesting thing about the Clone Wars is that in the films, in
the normal course of the Skywalker saga, it’s skipped over. You hear
about it, but you only see the very beginning and the very end of the
wars. But this was a turning point for the Republic. Star Wars: The
Clone Wars connects Episodes II and III. As a TV series, it gives
more depth and understanding about how Republic fell, and what a
fantastic hero Anakin Skywalker was. We got focused on him turning
into Darth Vader, that it’s easy to forget that he was one of the
biggest heroes of the entire Clone War. I think seeing him as a great
Jedi Knight reminds you that before it gets dark and tragic, Star Wars
is a fun adventure story, too.
With regards to the General Grievous / Anakin Skywalker continuity errors (i.e. them not knowing each other), I'm not aware of any time in the prequels they actually met. They may have been aware of each other, but that doesn't mean they'd recognise each other.
Of course, that doesn't mean there aren't other discrepancies in the series. Our friends over at the Sci Fi Stack have had a question on major continuity problems created by the series and responded with a few. To lift and shift them from the posted link:
"Since Episode IV, the Republic was known to have fought the Clone
Wars, and the EU added several other conflicts, such as the Great Sith
War; but in Episode II, it was stated that "there hasn't been a
full-scale war since the formation of the Republic." The
above-mentioned Ruusan Reformation also handled this discrepancy."
"Timothy Zahn, in his Thrawn trilogy, cast the Clone Masters as the
opponent of the Galactic Republic in the Clone Wars, as several
authors of the '90s considered that the cloners and their clones were
the "villains." However, after Attack of the Clones revealed the
clones were on the Republic's side, the Clone Masters were retconned
into being renegades who did not attack until after the formation of
"One example of a direct contradiction between The Clone Wars and
Clone Wars is the use of Eta-2 Actis-class light interceptors at the
same point in the timeline Anakin is knighted, whereas in The Clone
Wars Delta-7B Aethersprite-class light interceptors are used, which
were ostensibly used before the later Eta-2s. "
"For example, in the novelization, Anakin remarks that he has only
read about Grievous' Magnaguards in an intel report, despite having
fought them in the show."
"Another very noticeable continuity error is the fact that Anakin has
his scar which he received shortly after the Battle of Rendili from
Asajj Ventress. He then injured Ventress so severely she was forced to
recuperate until the Battle of Boz Pity, after which she deserts the
CIS and the war for good. However, as he was originally shown to
receive his scar in 19 BBY, yet the series takes place in 21 BBY, this
is a glaring continuity error."
Thanks to @Sam for these.
I think that ultimately, in a universe of this size, discrepancies are bound to occur. The largest, most noticeable ones were retconned. The others were likely not.