Let me put my bias right out on the table. I am a Park Sloper and have been for 30 years - before it became more well heeled. I do recall the time when Fifth Ave was drug infested and run down.
My first forays into WBurg were for the Brooklyn Brewery. Back when it opened, the scene had yet to develop and there was a great deserted vibe about the place. Like the Slope, WBurg has had a lot of changes. Here is a brief guide
Architecture - Here the Slope is clearly the winner. It is a far more unified vision and is a Historic District so you cannot destroy what makes it special. Mostly 3-4 story brownstones with apartments up by the park. Lots of mansions on 8th Ave and Prospect Park West. WBurg was more of a working man's neighborhood. The houses were always more modest. What WBurg does have is some great industrial buildings that have been repurposed. The problem is that development has placed lots of "sore thumb" buildings scattered throughout the area.
Food - You won't go hungry in either neighborhood. Both have their old school Italian faves [Bamonte's in WBurg and Two Toms and Monte's Venecian Room in Park Slope], both have their artisenal pizza [Motorino or Franny's]. Both have high end places [Peter Lugar or The Stone Park Cafe]. BUT - the food scene in WBurg may be the best in the city. Not only for restaurants but because they also have the Brooklyn Flea and Shmorgasburg.
Parks - Park Slope has Prospect Park - Olmstead's Masterpiece. WBurg has McCarren Park - a barren dirt patch infested by hipsters playing kick ball. But, they also have the new waterfront Parks including the Grand Ferry Park at the foot of Grand Street which is a little gem. But, seriously - you just cannot compete with Prospect Park and the nearby Botanic Gardens.
Stereotypes - The Slope used to be more known as the People's Republic for its lefty leanings and for the Park Slope Food Coop. Now, as money has moved in - you are more likely to hear the term "Stroller Mafia". Where overpriviledge meets overprotection. WBurg has the three H's - Hipsters, Hispanics and Hasids - but the tourist areas are know as hipster central. A sea of skinny jeans, ironic tee shirts, bad facial hair, and an attitude of trust fund self importance.
Drinks - First, with both Wburg and Park Slope anyone will be in beer heaven. Yes, both have their coctail and wine places, but I am a beer guy. I never go into Manhattan to drink anymore. My nabe of Park Slope has some great pubs. My faves include the Gate, High Dive, Mission Dolores, Loki Lounge and Union Hall. But, for my tastes, the pride of the nabe is Biercraft. Great Beer for sale or drinking on premises, free tastings on most Tuesdays and/or Thursdays, and growler refills. They now have a garden in back. Also Gourmet food to be washed down by the great beer, WBurg is also phenominal. Brooklyn Beer is made here, and you can sample from the source Friday night and Weekend afternoons. There is Loreley and Radegast Hall, both great biergardens, there are tons of great bars as well - Muggs, Spuyten Duyvil, Spike Hill, Union Pool Barcade and lots lots lots more. The scene is livelier and more varied in WBurg, but for nice peaceful pubs I prefer the Slope.
Nightlife. Park Slope has Southpaw, Union Hall and the Bell Jar. The Brooklyn Nets are coming in a year or two. WBurg has everything else. WBurg has taken over from the Village as the center for nightlife these days. This area comes alive after dark. Heck, the hipsters don't have jobs anyway so they do not wake uo before 4:00 anyway. After midnite, you still may not find a seat on the L train coming back from Bedford Ave.
Shopping - Both are well served. Park Slope has 5th Ave and all its boutiques lined up nicely. 7th Ave is more for traditional items and real estate brokerages. WBurg has lots and lot of trendy stores up and down Bedford and spilling out onto the sidestreets. But, as I am usually there to imbibe, as opposed to shopping, I really am not an expert in this area.
TA Personality: Park Slope - Main Proponent is Brooklyn Mel. Witty insightful posts with a modicum of snarkiness thrown in for that BK feel. Williamsburg [never call it Billyburg] has Lotus Path - famous for non-sequitors, video clips of monkeys and other useful travel advice. Where is NY Whiz when you need her?
WHERE SHOULD YOU GO. This is the easiest choice of all. BOTH. The strength of NYC is not it's skyscrapers, its museums, and the Times Square crush. The strength of NYC is its neighborhoods. And, these are two places that should be on any itinerary.