What is the difference between Mitigate and Mollify?

I have a feeling that mollify is used with anger and other emotions while mitigate can be used with severity of something such as punishment.

Mitigate: to lessen in severity or to appease

Mollify: to calm; soothe; make less severe

Am I right? If not than could someone explain the difference to me.

Mitigate:

verb

[WITH OBJECT]
1.0 Make (something bad) less severe, serious, or painful:

1.1 Lessen the gravity of (an offence or mistake):

The etymology suggests softening action:

early 15c., "relieve (pain),"
from Latin mitigatus, past participle of mitigare "soften, make tender, ripen, mellow, tame," figuratively, "make mild or gentle, pacify, soothe,"
ultimately from mitis "gentle, soft"
(from PIE *mei- "mild") + root of agere "do, make, act" (see act).
First element is from PIE root *mei- "soft, mild."

Mollify:

verb

[WITH OBJECT]
1.0 Appease the anger or anxiety of (someone):

1.1 rare Reduce the severity of (something):

The etymology suggests to make soft:

late 14c., "to soften (a substance),"
from Old French mollifier or directly from Late Latin mollificare "make soft, mollify"
from mollificus "softening,"
from Latin mollis "soft" (see melt (v.)) + root of facere "to make" (see factitious).
Transferred sense of "soften in temper, appease, pacify" is recorded from early 15c. Related: Mollified; mollifying.

In definition and connotation, these two words share a lot in common, but from the 16th century forward, the word mitigate has been applied more commonly to circumstances, while mollify has been applied more commonly to hearts and minds.