According to the Shulchan Aruch (O"C 131:4, 6-7) there are days on which we don't say Tachanun. In general, it seems that these are "happy" days -- holidays or communal celebrations. But while on those days we don't say Tachanun, we do say many other prayers which echo the same ideas.
On a standard Tuesday (assuming a minyan which doesn't say the 13 middot) the first part of the Tachanun is made up of selections from Nach admitting to sin and asking for forgiveness. We have already said these things in Shmoneh Esrei (along with the "Heal me" mentioned in the middle of Tehillim perek 6). We banged on our chests when admitting to sin and asked for mercy and salvation in the Brachot of Et Tzemach David and Sh'ma Koleinu.
The second part of Tachanun, Shomer Yisra'el, uses the language said in the Sh'ma to speak about Hashem's role in guarding us (reminiscent of the Shin Pasuk from Ashrei).
The third section, using a variety of Pesukim, reminds us that we cannot stand on our own merit, but that we have to look towards Hashem, similar to what we say after the initial Brachot in davening, "Ribon kol ha'olamim, lo al tzidkateinu anachnu mapilim..."
Davening has repeated motifs including our reliance on Hashem, or recognition that we have sinned and a realization that we need to be guided and guarded. Why is the Tachanun prayer, which has statements found elsewhere not skipped, skipped on a happy day (when we don't need guiding, guarding and humility?) ?
What is essential to skip and then, shouldn't we still say the other parts, as we already have in davening?
Note -- this is an offshoot of Why Don't we Say Tachanun on a Particular Day? about the specific content of Tachanun, not the decision to say it or not.