By Davan 
Why does it seem like Clemenza in The Godfather was more trusted with important jobs, like hiding the gun for Michael and divising the plan how to shoot Sollozzo? Also showing Michael how to handle the gun, and Tessio seemed to be in the background?
It’s a good question and a lot of the answer seems to be demonstrated in the scene when we see Clemenza fondly teasing Michael about Kay and then teaches Michael how to make a big tub of pasta for a crew that’s gone to the mattresses.
“Hey, Mikey. Why don’t you tell that nice girl you love her? I love you with all o’ my heart! If I don’t see you again a-soon, I’m-a gonna die! Come over here, kid. Learn somethin’. You never know. You might have to cook for 20 guys someday. You see? You start out with a little bit of oil. Then you fry some garlic. Then you throw in some tomatoes, some tomato paste. You fry it. You make sure it doesn’t stick. When you got it to a boil, you shove in all your sausage and your meatballs.” [Clemenza to Michael after Sonny has sent everyone to the mattresses.]The Godfather (1972)
It seems like Clemenza has more important jobs mostly because we see him more often than Tessio than for any other reason. But, despite his eventual duplicity, Tessio had a role no less important than Clemenza in the rise of the Corleone Family to the top of the New York underworld. But there are indications Clemenza was more of a favorite than Tessio among the Corleones, no question.
Even in his appearance at the beginning of the film, when he’s dancing to the old traditional Italian wedding music before he scolds Paulie to do his job, Pete Clemenza just seemed to be looser and a lot more jovial than his old friend and compadre, Salvatore Tessio. This was also demonstrated in the novel by the fact that Vito asked Clemenza to be godfather to Santino, implying Clemenza spent more time with the Corleone Family in his down time than Tessio, and it is Clemenza, rather than Tessio, who trained Sonny when the eldest Corleone son entered the business.
There’s also an implication based on the initial confidence Clemenza put in younger Vito by asking Vito to hide the firearms the police were presumably looking for when they first met in The Godfather Part II, that Vito and Clemenza established a stronger bond than Vito did with Tessio, simply by Vito having kept Clemenza’s confidence upon one request and with no questions asked, suggesting Clemenza had a good nose for loyalty and returned such loyalty tenfold, although until the end, Tessio’s loyalty to Vito was stalwart in its own right. It was only when Barzini started eating into Tessio’s territory and Vito handed the reins over to Michael that Tessio began to negotiate his treachery.
All that said, I’m not sure it’s necessarily that Clemenza was given more important jobs, as much as Tessio’s forte was in dealing with the Corleone business in the field and managing assets out among the Corleone territories in New York. Tessio was sort of the eyes and ears on the street, while Clemenza was more adept at managing the inner gears of the Corleone machine. Clemenza had his ear to the rail as well, with his own people out and about, but we see more of Clemenza because he was more often required to perform activities within eyeshot, whereas Tessio was much more invisible by way of physical distance, but no less critical to the success of the Corleone Syndicate until his ultimate betrayal of Michael.
In fact, being out and about more probably lent itself to Tessio being more susceptible to being turned, just being more exposed and more easily influenced by the likes of Emilio Barzini.
Clemenza simply always remained more loyal to the Corleones.
