Agree with this. Don't expect much for service, but the rooms have a lot of character (try to get one with a balcony overlooking the creek), the area in Dubai's old town is interesting, and the rates should be well within your budget or even below.

A few other options:
-- Conrad. This should be doable for your price and is a solid upscale hotel centrally located. I like walking on SZR, though it's not that interesting for a first-time tourist. Still, there's a metro stop right in front and easy access for taxis. This is probably where I would stay given your preferences if not the Al Seef Heritage.
-- Hilton Palm. This is a decent new resort with good on-site amenities and excellent Diamond treatment -- I got a room upgrade, lounge access, breakfast including a special "Diamond menu" that had some thought put into it, and some drink vouchers. You can walk along the corniche/beach, with not a lot to see. It's also difficult to get on and off the Palm.
-- Hilton Dubai Creek. Avoid. This project seems ill-advised. The hotel has no driveway, no bell service, and no real lobby, it's bizarrely only accessible by foot or through the huge underground parking lot, entering at the Marriott Marquis on the other end of the Jewel of the Creek (or maybe it's Pearl of the Creek) complex. They told me to leave my car at the Marriott and take the buggy (after waiting for the buggy to come from the Hilton, which is surprisingly far away). This is an apartment hotel and does have a pool. It was my fault for not paying close attention, and the rooms are decent enough for what they are, but the hotel has few amenities, no bar, and only a basic all-day-dining restaurant. It really should not be branded as a full-service property, but rather as Homewood Suites or the like (this brand doesn't exist in Dubai).
-- Habtoor City. This place is criticized here, but I found it decent. The rooms are nice and there are a lot of amenities on property. There isn't a lot to walk to in the immediate environs surrounding the hotel, but it's centrally located. I would give it a look (though it may not be a Hilton by next summer). The V Hotel Curio Collection is in the same complex, but I'd avoid it. It's a rebranded (very short-lived) W, but really tacky -- they kept some of the same W signage (the "V" is the same font as "W," like they just cut the letters in half) and, although it maintains some design elements of the W, it has a feel of a decaying hotel that wasn't up to brand standards so was reflagged half-heartedly.
-- Embassy Suites Business Bay. This is scheduled to open later this year as the first Embassy Suites outside North America I believe, and looks much nicer than a typical Embassy Suites (with rates to match). Again centrally located. I look forward to giving it a try when it opens. The DoubleTree Business Bay nearby is also surprisingly nice and well-located. I'd avoid all of the other DoubleTrees in the city.
-- Hilton or Hilton the Walk, JBR. I don't like this area. Although it's lively, you could be anywhere in the world. The Hilton is dated and traffic getting in and out of this area is bad, though summer will be a bit better.