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The accepted answer to this other question mentions that Mormons have two types of meeting locations: temples and churches.
According to the answer, the churches have a 3-hour meeting every Sunday.
In what kind of cases or times would a Mormon then visit a temple for a meeting?
It's not a choice between the two as the question seems to be asking. The temple is not actually a meetinghouse where ordinary worship services are held. (In fact, temples are closed on Sundays.) Latter-Day Saints regard the temple as the House of the Lord, a highly sacred place where, like the temple in the Old Testament, very special, sacred ordinances are performed.
The majority of the work that takes place in temples has to do with the LDS doctrine of salvation for the dead. Mormons believe that those who died without the Gospel can accept it in the hereafter, as has been discussed elsewhere on this site, but also that certain saving ordinances, such as baptism and confirmation, are necessary for salvation. This apparent dilemma is resolved by proxy work: in the temple, people can do this work on behalf of those who have passed on, consistent with Paul's mention in 1 Corinthians of baptism for the dead. As far as I'm aware, ordinances on behalf of the dead cannot be performed anywhere but inside a temple.
One does not replace the other: a faithful member of the Church will attend regular, weekly Sunday meetings and attend the temple when possible. Temples are usually open during the week and Saturday, as opposed to Sundays. The ordinances of the temple do not replace the ordinance of the Sacrament administered in Sunday meetings, and both temple and Sacrament ordinances are necessary for salvation and exaltation.
Some members attend the temple regularly; others go when they can, or when they need to. But it is encouraged by Church leaders to attend the temple often and always hold a current temple recommend (certifying one's worthiness) to be able to enter it.
More information about temples, including their schedules, can be found here: http://temples.lds.org/
The reasons a Mormon goes to a temple instead of a church can be found in Preparing to Enter the Holy Temple manual:
In the Church we build buildings of many kinds. In them we worship, we teach, we find recreation, we organize. We can organize stakes and wards and missions and quorums and Relief Societies in these buildings or even in rented halls. But, when we organize families according to the order that the Lord has revealed, we organize them in the temples. Temple marriage, that sealing ordinance, is a crowning blessing that you may claim in the holy temple.
In the temples members of the Church who make themselves eligible can participate in the most exalted of the redeeming ordinances that have been revealed to mankind. There, in a sacred ceremony, an individual may be washed and anointed and instructed and endowed and sealed. And when we have received these blessings for ourselves, we may officiate for those who have died without having had the same opportunity. In the temples the sacred ordinances are performed for the living and for the dead alike.
An analogy would be asking "When would a Jew in the first century go to the Temple in Jerusalem instead of the local synagogue?" They would go to the local synagogue on the Sabbath to meet with their congregation and less frequently to the Temple (typically for festivals like Passover) to participate in rituals.
Latter-day Saints go to their local chapel on the Sabbath to meet with their congregation. They go to the Temple less frequently to participate in rituals. How often is up to individual members. Some go 1+ times a week, 1+ times a month, 1+ times a year, maybe only for things like weddings, maybe not at all.
There isn't really any overlap between the types of meetings that are held at chapels and the types of rituals participated in at the Temple. Basically, this is similar to the difference between what was done in a synagogue versus in the Temple in 1st century Judea. For example, animals were only sacrificed at the Temple in Jerusalem while in the synagogue they were primarily doing things like hearing and discussing the scriptures.
