It seems to me that, while one can manage his/her time outside of a project, it would be difficult to manage a project without managing time.

I could not imagine a project with a set of tasks, and no dates.  How can one plan without time references?

On the other end of that spectrum, if a project is late, can it still be considered successful? In some industries, being late is part of the deal, it's normal and expected. However, in other businesses, being late mean penalties and a tarnished reputation.

So, what is the most important: doing it right, or doing it on time? 

My position here would be both. It's no use delivering something on time, if it's not fully completed and functionnal. There is nothing worse for a comapny's reputation than a beta-level product released. Clients will forgive lateness to a certain extent, but they will not forgive and product that doesn't work.

However, there is a limit to a client's patience. Sometimes, when trying no to make it right, but to make it perfect, we miss the window of opportunity for the product. Either another company beats you to market, or the need for the product has disappeared. Just think of the company trying to develop the perfect BETA tape...

We have to do it right, and not too late.