The way I like to put it is that its like the difference between a two-wheel-drive vehicle and a four-wheel-drive vehicle.
Goenka's method, (U Ba Khin's method) lays more emphasis on mindfulness of respiration (which is mindfulness of the body), and mindfulness of feelings, than it does on the other two foundations of mindfulness. From my recollection (I undertook numerous 10-day retreats before my ordination in 1979), there is little mention of mindfulness of consciousness (cittānupassanā satipaṭṭhāna) and mental objects (dhammānupassanā satipaṭṭhāna).
For the raw beginner, the Venerable Mahāsi Sayādaw's method also stresses mindfulness of the body as the primary object (abdominal movements and bodily movements while walking etc.), but very soon the meditator is asked to extend his/her awareness
to all four foundations of mindfulness.
A two-wheel drive vehicle is fine on a smooth road in ice-free conditions, but a four wheel drive vehicle is much better off-road or in snow or in difficult driving conditions.
Similarly, Goenka's method is fine in a protected retreat environment, but hard to maintain and develop in daily life. It seems that many meditators struggle with this conflict, unable to practice effectively outside the confines of the meditation centre. I know this is the case as several of my supporters have attended these retreats.
Technically, the U Ba Khin method is Samatha-Vipassanā, while the Mahāsi method is Suddha-vipassanā — that is the meditators strive to develop insight from the very beginning of practice, as soon as some momentary concentration has been established. Meditators on Goenka's courses do three days of samatha, then switch to what they call "vipassanā" on the fourth day, when they start sweeping around the body to observe sensations.
See also, Venerable Ledi Sayādaw's Manual of Resipiration
on how to proceed from concentration to insight. He is the root teacher of that tradition.
The biggest fault with the U Ba Khin method is the lack of any systematic walking practice. There is too much emphasis on sitting. The result is a tendency to sloth and torpor, a higher likelihood of "special experiences" that may mislead the unwary meditator (see The Ten Corruptions of Insight
, and an over-sensitivity to suffering.
The biggest fault with the Mahāsi method is that the meditator who doesn't practise diligently or lacks a skilled meditation teacher, may misunderstand the method, and think too much due to lack of tranquillity. Effort must be constant, diligent, and continuous throughout the entire day without a break.
The Buddha's method is what we should all practice. Strictly speaking it is not vipassanā meditation but mindfulness meditation. If we establish both mindfulness and concentration, insight will arise — otherwise it won't whatever we call our meditation practice or technique.
The essential point is to be mindful of each and every mental and physical phenomenon as it arises in the present moment. The particular technique that you use to get to that point is not the issue. Just use whatever works for you — Sunlun Sayādaw developed insight while hoeing his fields when he was still a farmer. Afterwards he became a monk, and his followers claim that he was an Arahant.