A three head tape deck theoretically will have better performance, in terms of top end frequency response, than a two head tape deck because the head gap width requirements are not the same for recording and playback functions. A two-head deck has a combined record/play head and an erase head. A three-head deck has separate heads for recording and playback, plus the erase head. In a two-head deck, the gap in the head is by necessity a compromise between what would be the head best gap width for a recording head and what head gap width gives the best playback response. So this would lead to the conclusion that a three-head deck is always better, but this is not always true as we shall explore.
By comparison, if we look at a reel to reel tape deck, all of the tape transport parts and especially the tape path is much larger than for a cassette deck. This gives a big advantage when trying to construct a machine with separate heads for recording and playback. There is plenty of room to place a separate erase head, followed by the recording head, followed by the playback head in the tape path between the two capstan/pinch rollers that are used in this type of closed loop system. Here too tolerances have to be very good for proper alignment between these heads. On many reel to reel decks, each record and playback head has its own azimuth adjustment so that it can be finely aligned once installed in the tape deck.
In the case of a cassette machine, everything is much smaller and the design of the cassette itself is limiting in that the record/playback head must fit into a very small opening in the center of the cassette shell. The erase head fits into its own opening to one side of the record/playback head. There is not nearly enough room to fit the typical separate record and playback heads together inside of the cassette shell, so originally this limited cassette decks to just two heads.
Eventually the Japanese, I believe it was Nakamichi, engineered a true three head cassette deck by building a deck with miniature separate recording and playback heads. This design was very expensive to manufacture, and also created the problem of getting/keeping good alignment between the recording and playback heads.
Most of the three head decks that followed solved this by packaging the separate miniature record and playback heads into a single housing, with a fixed azimuth alignment between the record and playback heads. This was both good and bad. Good in that if the heads were aligned correctly during manufacturing, then they would always perform optimumly. Bad in that if the heads were manufactured even a tiny bit out of alignment (cassette deck head azimuth alignment tolerances are unbelievably tight for proper performance to be achieved), that record/playback performance would always be compromised, with no way to correct this misalignment.
A two-head deck does not have this alignment problem because the record head is also the playback head. As long as this one head is aligned with the tape, performance problems due to record/playback azimuth alignment errors are not a factor.
In the case of the cassette tape recording, the results you get are very much dependent on the quality of the cassette shell mechanism as well as the tape. Also, the quality of the parts used in the deck and the manufacturers quality control has more to do with the results than any one design approach. There were some very good two-head decks and some very bad three head decks. There were also some very good three head decks, however, these were typically much more expensive tape decks.
The original audio cassette was designed by Phillips for use as a portable dictation device, or for recording lectures. As soon as people bought the first Norelco cassette player/recorders they proceeded to make cassette recordings of their favorite music to take along on trips etc.
The trip from being a low-fi portable dictation medium to challenging the vinly LP for the consumer market lasted 10 or 12 years. Along the way one estimate stated that during this same period the Japanese tape and electronics industry spent the equivalent of what the U.S. spent on the Apollo program to make the audio cassette a true music medium.
The cassette never did overtake the LP, but it killed off the 8-track and also was the death of reel to reel tape for the consumer market. By the early 1970s, using type II tape and Dolby B noise reduction, the typical consumer deck could make a copy of an LP that to most people was identical to the record and was portable.