Old Capulet does not allow Tybalt to confront Romeo because such an action would challenge the older man's position as head of his household. Furthermore, such action at "an old-accustomed feast" would be deemed an insult to the guests. As is evident in the text, the houses of Montague and Capulet are the leading influences in Verona, and should try to avoid scandals.
Old Capulet doesn't allow Tybalt to confront Romeo because Capulet (and Montague) has been scolded by the Prince to stop fighting because it causes a big scene and puts others in danger. The prince also has talked to the both of them three times so Capulet knew he would face consequences if he allowed his servant to attack a Montague.
In order to answer this question effectively you need to review what occurred in the previous scene. After the dispute between Montague and Capulet a warning was issued by the Prince to both families anyone who started another fight would forfeit their lives. This is ONE of the motivating factors.
Another factor is that the dispute would be in his house. He did not want the violence to occur at his place and so he bid Tybalt to halt his agression.
When Tybalt made the move to do this anyway stating that he could not "bear it". Capulet saw this as a threat to his power and influence. Tybalt sees himself as the heir apparent and Capulet sees Tybalt as a threat to his power and influence. If he cannot control a member of his own house then he has lost face within the community. Capulets power base is relient upon power and influence which Tybalt threatens through his actions. Not only the power within his own family but the power within the community including the favour of the Prince.
As we have found out at the beginning of the play, if the Montagues and Capulets fight one more time, they will be killed.
So, when Romeo Montague showed up at the Capulet's ball, Lord Capulet thought of what Prince has said and let Romeo stay and enjoy the party. What Lord Capulet didn't know at the time was that party, his only daughter; Juliet will fall in love with the young Montague heir. So letting Romeo stay has started a part of the many parts of the tragedy of Romeo And Juliet
With Love and Care,
He felt all these actions that Tybalt did would make him lose his influence and control over the household and make him lose face and honour. It would also challenge his position and rightful standing and maybe even be forced down.
Such actions may prove to be an insult and a disgrace to the guests and even shamed Capulet. They might be rumors saying that Capulets are very aggressive and violet, which would make him lose his power and his might among the people.
The Capulets are the main strongforce in this town and one wrong action or move might be lead to a full-blown conspiracy and scandals might break out. They should be trying to avoid scandals and not creating them.