What does Emilia's remark about the rift between Othello and Cassio suggest about their relationship?

I'm not exactly sure what quote you mean.  Is it this one
perhaps?

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Good morrow, good Lieutenant: I am
sorry
For your displeasure; but all will sure be
well.
The general and his wife are talking of
it;
And she speaks for you stoutly: the Moor
replies,
That he you hurt is of great fame in
Cyprus,
And great affinity, and that in wholesome
wisdom
He might not but refuse you; but he protests he
loves you
And needs no other suitor but his
likings
To take the safest occasion by the
front
To bring you in
again.

In
Othello Act III, Emilia summarizes the closed-door "trial" of
Cassio here.  Just as Othello battled Brabantio in court in Act I and won, so too does
he battle Desdemona and win regarding the plight of
Cassio.

Emilia hopes that her husband too will be a kind of
"lawyer" in the matter:

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I warrant it grieves my
husband,

As if the case were
his.

Desdemona tries to argue that
Othello loves Cassio, but Othello responds that he must protect the status and
reputation of the wounded man, Montano.  In his decision-making, Othello values
reputation above "love."  In short, Cassio wounded Montano AND Othello more in terms of
reputation than in bodily harm.  He must, therefore, be punished by a demotion in rank
and reputation.

Later, after Emilia steals the
handkerchief, she will tell Desdemona:

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'Tis not a year or two shows us a
man:
They are all but stomachs, and we all but
food;
To eat us hungerly, and when they are
full,
They belch us. Look you, Cassio and my
husband!

After Iago
has used her for getting the handkerchief, Emilia will lose her optimism that either
Desdemona or Iago can talk Othello into forgiveness.  She believes that men are
consumers only, of women and men of lower status.

It is
clear that Emilia's remarks show that Othello values his public reputation above
personal relationships.