Which of the verbs discriminate/differ or any other similar verb seems more natural and professional in this sentence?
We focus on the approach itself, why it is an appropriate solution, and how it discriminates/differs from other solutions.
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As Tᴚoɯɐuo mentions in his comment, you use "discriminate" incorrectly in your example.
Discriminate (transitive verb): to mark or perceive the distinguishing or peculiar features of : to distinguish by discerning or exposing differences : to recognize or identify as separate and distinct
Something can discriminate between two things, for example:
His tastes were supposedly so refined that, from a single sip, he could discriminate between whiskeys from different Scottish distilleries.
"Discriminate" is not synonymous with "differ". "Differ" means "to be different from," while discriminate meets "to distinguish the difference between."
Although they say these two whiskeys differ, I can't discriminate between them.
In your example "differ" would be correct, although you can change it to use "discriminate":
We focus on the approach itself, why it is an appropriate solution, and how you can discriminate our solution from other solutions.
However "distinguish" sounds better than "discriminate" here, so I wouldn't recommend this.
Note also that "discriminate" has a secondary, often negative, meaning:
Discriminate (intransitive verb): to make a difference in treatment or favor on a basis other than individual merit
In the United States it is illegal for most businesses to discriminate on the basis of race, gender, religion, or ethnicity. In most states it is also illegal to discriminate on the basis of age, marital status, sexual orientation, and physical ability.
The CEO of the company was recently forced to resign amidst accusations that he discriminates against women.
Because discriminate has this second, negative meaning, you have to be careful not to imply the wrong thing when using it. For this reason other words like "distinguish" or "differentiate" may be better.