Any answer is inevitably going to be patchy. Our knowledge of the seventh century is much poorer than what we have for the twelfth. We are much better informed about religious foundations than the demography of housing, for example. For a general and highly readable account of the city at the end of the twelfth century (although using many elements from earlier for the sake of illumination) by a a scholar in the field see Harris, Jonathan. Constantinople: Capital of Byzantium. London: Continuum, 2007.

Between these two dates, the city will have changed in a number of ways, visually, at least. The degree to what effect these changes had on patterns of life in the city in a whole different topic. A visitor in 659 probably would think that the population of the city was much too small for its size. We do not have demographic figures, but in 618 the government ended the free bread dole, as the loss of Egypt to the Persians had made it unsustainable (Chronicon Paschale, anno 618). The Avars had also cut the main aqueduct in 626, which was only restored in the eighth century (Theophanes, Chronographia, AM 6258). By the twelfth century the population had rebounded. Although this again by a reflection of our patchy source material, you would probably see a larger variety of foreigners, particularly those from the medieval west, in the twelfth century. The establishment of a Muslim quarter is hard to judge, but there was certainly was one after the 1180s, and there is good evidence for Muslims coming to trade in the city from the tenth century (Reinert, Stephen W. “The Muslim Presence in Constantinople, 9th-15th Centuries: Some Preliminary Observations.” In Studies on the Internal Diaspora of the Byzantine Empire, edited by Hélène Ahrweiler and Angeliki E. Laiou. Washington, D.C.: Dumbarton Oaks, 1998, 137-142.)

A concern about violence would not have been far from the mind of someone living in 659. Constantinople had been besieged by the Arabs in 654 (following Pseudo-Sebeos rather than the Greek tradition, see O’Sullivan, Shaun. “Sebeos’ Account of an Arab Attack on Constantinople in 654.” Byzantine and Modern Greek Studies 28, no. 1 (2004): 67–88.) The city had also been besieged by the Avars in 626 and would again be besieged in 669 by the Arabs (for the new dating, see (Jankowiak, Marek. “The First Arab Siege of Constantinople.” Travaux et Mémoires 17 (2013): 237–320.) The people in 659 would be seeing their emperor making preparations to depart the city, as Constans II spent 660 in the Caucasus rallying allies. This is in stark contrast to 1167. Although the passage of the Second Crusade had certainly aroused fears and was within living memory, the city was secure and had not faced an existential threat probably since the last big Rus' naval raid of the mid-eleventh century. Perhaps the Sicilians firing arrows into the palace ca. 1153 from ships counts, but I wouldn't tend to say so (Niketas Choniates, trans. Magoulias, p. 57).

In terms of buildings, a lot changed. The praitorion in 659 did not yet have a mosque for visiting Muslim dignitaries. Justinian's church of Hagia Eirene (near Hagia Sophia, now inside the Topkapi walls) had been destroyed in an earthquake and the present structure was rebuilt in the eighth century. One of the more visible changes would have been the Monastery of Christ Pantokrator, a Komnenian foundation. The topography of imperial power had shifted. In the seventh century it was based in the imperial palace near the Hagia Sophia, whereas in the twelfth century the house of Komnenos spent a lot of time in the Blachernai quarter in the NW of the city.

You may also want to check out Byzantium 1200, a very good visual reconstruction of the city at the end of the twelfth century.: http://www.byzantium1200.com/