As regards the Catholic Church, the Mass that is now in common use—the Mass promulgated in 1970, sometimes referred to as the “Ordinary Form”—prescribes kneeling only during the Eucharistic Prayer, during which (in accord with Catholic belief) the bread and wine are transformed into the Body and Blood of Jesus Christ.
The faithful kneel at the so-called epiclesis (that is, the invocation of the Holy Spirit, asking Him to transform the gifts of bread and wine), and rise just before the priest says Mysterium fidei (“The mystery of faith”).
At the beginning of the Mass, however, the faithful are already standing, and they remain standing until the end of the collect, or opening prayer, at which time they sit to listen to the readings.
All of the instructions can found in the General Instruction of the Roman Missal (GIRM). See especially number 43.
Note that in many English-speaking countries, the faithful kneel from the end of the Sanctus (the Holy, Holy Holy), until either the Mysterium fidei or after the great Amen (that is, the end of the Eucharistic Prayer). The edition of the GIRM cited is for the United States and explains this latter practice. In the United States, in most dioceses the faithful also kneel for the Agnus Dei (Lamb of God).
(See the GIRM in Latin for the universal norms.)
As regards the Anglican liturgy, it is largely derived from the Catholic Roman Rite, but it developed long before the Mass that was promulgated in 1970, and so retains elements of the pre-1970 Mass (or “Extraordinary Form,” as it is called nowadays). Since the pre-1970 Mass does not prescibe specific liturgical postures for the faithful, those who adapted the Mass for the Anglican Book of Common Prayer made slightly different decisions as to where the kneeling should be. (See, for example, a parish communion guide from 1940.)
In the Catholic Church (specifically in the Western Church, because the symbolism in the Eastern Churches is different), kneeling represents profound respect and especially adoration. In the context of the liturgy (both the Mass and the Liturgy of the Hours) it is limited to the key moments in which God is worshipped; at Mass, the consecration and (in the United States) at the “Lamb of God.” Standing is already a posture of respect and prayer—which is why it prevails during most of the Mass—but kneeling is reserved for the most intense moments of adoration. That is why, for example, kneeling would not be appropriate during the priest’s presidential prayers (of which the opening prayer, or collect, is one). (There is an excellent short overview of the different postures at Mass by Fr. Edward McNamara, L.C.)
Why do the Anglicans kneel during their opening prayer? I expect that there are fundamentally two reasons (although I would invite a member of the Church of England to expand on this answer). One is that, as I mentioned, the Anglican liturgy is adapted largely from the older pre-1970 Mass, in which it was common (but not exactly mandated) for the faithful to kneel during nearly the entire Mass. The other is that those who composed the Book of Common Prayer had mostly a Protestant outlook: specifically, they tended to reject the sacrificial aspect of the Eucharist, as well as the doctrine of Transubstantiation (the transformation of the bread and wine into Jesus). Hence, it was logical that they should place less emphasis than Catholics do on the consecration of the Eucharist—they simply did not think of the Consecration as a more “intense” moment of adoration, the way Catholics would.