Besides having the extra square footage--and a kitchen if you are inclined to cook local produce for yourself, linger over coffee in the morning, or at least keep snacks and drinks handy--staying in a flat is the best way to get to see native life close-up. You'll walk the same paths to the tram as the locals, you'll shop where they shop, get bread at the neighborhood bakery...a much more intense experience of the sense of place than you can sometimes get from a hotel. If you need room service or a concierge, a rental is probably not for you; but if you are more independent, give it a try sometime. Over the years I have developed some great resources and have always been satisfied with the places we've rented, and have researched the various neighborhoods' characteristics in advance (which allows us to choose wisely). 
In Istanbul we chose the other side of the Golden Horn--lively Beyoglu. It's more modern (that term being relative, in that modern means sometime after the 13th c.) and definitely more European in feel. Some of the early settlers were Roman Catholic Genovese traders, who were trying to steer clear of the Eastern Orthodox church. (They're responsible for the tower that dominates that peninsula.) After the Ottoman conquest, Beyoglu became the home of European ambassadors and such, and the merchants with foreign roots wanted to be near their embassies, so this "side across" became quite prosperous and genteel. Later, other non-Muslim groups drifted over from the older Istanbul as well--Greeks, Armenians, and Jews all made their homes here in the 18th and 19th centuries. As a result, Beyoglu looks and feels very European; there were many times I felt I could just as easily have been walking around Paris (but with hills). Buskers and street vendors abound (you're never far from a simit or roasted chestnuts).
Our apartment was along the back side of the Galatasaray Lisesi (high school), whose elaborate gates dominate the peak of Istiklal Caddesi, the 24/7 throbbing main artery of Beyoglu that runs from the charming Galata end up to bustling Taksim.
Istiklal and its side streets are lined with first floor shops, cafes, restaurants, banks, etc; in short, everything you would need; and on the upper floors are flats (or in some cases, on Istiklal itself, probably offices). In the narrow streets behind the Çiçek Pasaji you will find a small market with fresh seafood, meat, spice and produce stalls lining the way. We shopped here for dinners sometimes. I chose this neighborhood--as opposed to the area closer to some of the major historic and tourist attractions--because it would have good options for dining and we'd be close to home in the evening.
It's very easy to hop the Tünel and Tramway to get to those places, as well as to the various ferry docks, or even to walk across the Galata bridge with its hundreds of fisherfolk. 
At the Galata end you'll find a street full of music shops (Galip Dede) and another lined with electrical shops. This kind of organizations makes an urban dweller's life easy: you need a lamp, you go to that street; you need a faucet, you go to the street with all the plumbing supply shops. You want antiques or second-hand furniture and chatchkes, you go to Cukurcuma behind the Galatasaray Lisesi.
Attractions on this side include the Pera Museum and the Mevlevi Lodge, where, on the right day, you may be able to get into see the dervishes do their thing--and this is the only REAL dervish session in Istanbul, the others are not really religious in nature. There's also a small and informative Jewish Museum near the Karaköy end of the Tünel.
I'm posting some "street life" pictures today, like the cobblestone repairmen in front of our building...and this fellow at the Hippodrome with his scale. I saw another man with a scale in Galip Dede St. and then this second one across the Golden Horn. Maybe lots of folks don't have scales at home? Seemingly there is some market, no matter how small, for people to pay a few pennies to weigh themselves??? 
Anyway, I had to take a picture of this scale man because he was so friendly. I was at quite a distance, using a 10x zoom lens on my digicam and he still figured out I was shooting him...unless he was waving at someone else!? Fortunately I could be more circumspect when I captured the street workers resetting the cobblestones of Turnaçibasi Sokak.
