Seattle is a larger city, which has districts that are fairly different in style. That's a whole other level we can get into later. In general though, it's a much more active place (than Bellevue), with a lot more people and a lot more things to do. It's also dirtier, and has a non-insignificant homeless population which can be scary if you're riding the bus alone late at night (or worse, at 2:30 on a workday). Worth noting that they're not normally dangerous, but it an be unnerving, and you may get harassed (as in any big city). It's farther from work too, which is a definite downside if you don't have a car, as the bus can take up to 25-30 minutes to get to work. All in all, I still prefer the big city due to the amount there is to do, the types of people you're able to meet, and the strong prevalence of fresh-out-of-college tech employees in the area. However, that's not everyone's gig and I know that.
Disclaimer: I'm not listing every neighborhood here. Some, including Wallingford, Madison Park, North Queen Anne, and Laurelhurst have been left out. This is not because they're bad places to live, but because I have a limited amount of typing ability and they are less-lived in than the ones I did mention (or they're very similar to the neighborhoods near them, like Laurelhurst).
This district gets it's name from the University of Washington, which is located right in the heart of UDist.
The red outlined area is the UDist. The yellow bar next to 15th ave is "The Ave"
UDist is close to Gas Works park, and contains the University of Washington (aka "U-Dub", no joke that's what people call it) campus, which is beautiful and a lot of fun to walk around. Just to the left of UDist is University Village, an absolutely beautiful outdoor shopping and dining area which attracts a lot of business, especially in the summer time.
For other attractions, there is "The Ave", which is University Way NE. It contains most-everything a college town would contain, including food, a movie theater, and a ton of Indian, Japanese, Chinese, and Korean food. There are also plenty of places serving American food, so don't freak out if the international cuisine is not your forte.
UDist is about an hour walk from downtown, but you can take a bus there in about 10-15 minutes. One thing worth noting is that parking at these apartments is definitely cheaper than downtown; it's only about $125-$150/month. Still a lot if you ask me. The light rail was recently extended from the UW Stadium to a new station in UDist and up to Roosevelt and Northgate, and there are lots of buses, which makes getting around very easy. The 542 bus line, which goes to Redmond, runs through UDist, making for a good commute.
If you're looking to buy, UDist is a solid location as long as you are okay with the college town vibe. There is some crime and a non-zero homeless population, but comparable to other areas of the city.
The people here tend to be college students. This is worth noting, because it means that your neighbors in your apartment building are likely to, well, be college students. This may include parties and late-night hangouts, which can be loud. Little-known tip is that going farther north on The Ave moves you away from that crowd, and you can get apartments with mostly working professionals.
This district surrounds Green Lake, a man-made lake hosting swimming, fishing, and boating.
The lake in the center of the red outline is Green Lake, with parks next to it on either side.
Greenlake is just off I-5, giving it fairly quick access to much of the city by both bus and car. It has its own little shopping area east of Green Lake Park, and a larger one on the other side of I-5. It's an extremely popular location for families to go swimming, and the beach in Green Lake Park is often very crowded. There are a number of Microsoft employees living in that area too.
Greenlake is about an hour and a half walk from downtown, and the bus is also longer than UDist: about 25 minutes. Parking is definitely cheaper here, around $75-$100/month, but it's worth noting that the rent in this area is inflated to hell. It's a very popular area right now, and tends to emulate Bellevue in many ways. It's a cleaner area of town, and has a semi suburb-y vibe. This also leads to prices going up. Food here is going to be similar to downtown in terms of price, as will most attractions.