What are the differences between a DHD and Earth's dialing computer?

Our program is nowhere near as sophisticated as the one that exists in the DHDs. It's completely jerry-rigged.

—Samantha Carter, "Avenger 2.0"

First off, the pilot Children of the Gods establishes that the DHD was missing from the Giza site, so Earth had to create their own. It wasn't until one was found with the Antarctic gate in "Solitudes" that they could study one on Earth. Among the differences, Earth's dialing computer:

• Stores addresses to planets, whereas a DHD is just an input device (i.e. you have to memorize the addresses).
• Initially was a rough ride, causing people going through it to tumble out on the other side and have ice crystals appear on them. (Stargate film, "Children of the Gods")
• Initially caused the gate to shake a whole lot (Stargate film, "Children of the Gods") until inertial dampeners were added. (Referenced in "Solitudes")
• Does calculations for stellar drift based on the Abydos cartouche. ("Children of the Gods")
• Could not dial the eighth chevron until O'Neill, with the Ancient repository in his head, upgraded the computer to dial the Asgard's home galaxy. ("The Fifth Race")
• Dials slower than a DHD, but by ignoring a number of checks can use Sam's "fast-dialing" algorithm to dial faster than a DHD. ("Serpent's Song")
• Requires palm scanners to operate, making it impossible for races like the Reetou to use it. ("Show and Tell")
• Periodically "cold dials" planets that previously could not establish a wormhole, thus allowing exploration if they some day become accessible again. ("New Ground")
• Allows one to disable safety protocols when trying to establish a lock on a gate, which once resulted in a wormhole passing through a sun and damaging it. ("Red Sky")
• Ignores 220 of the 400 feedback signals the Stargate can give when it dials. ("48 Hours")
• Does not receive data from the Stargate network's correlative update service, making it immune to efforts to tamper with it, such as the Avenger virus. ("Avenger 2.0")
• Allows you to easily recall recently dialed addresses (getting it from a DHD recalls taking it apart and examining a control crystal imprint with special equipment, as seen in Stargate Atlantis' "The Lost Boys", or using Adria's unique ability from "The Quest, Part 2")

Additionally, the same technology as the dialing computer was used in Stargate Atlantis for the McKay/Carter Intergalactic Gate Bridge. A macro that McKay wrote allowed Stargates to store a person in a buffer and be forwarded from one gate to another without rematerializing. ("The Return, Part 1") A workaround was written to prevent a Pegasus gate from always taking priority over a Milky Way gate. ("Midway") This was not possible with an unmodified DHD.

Note that in the Stargate Universe pilot "Air, Part 1", Icarus Base had access to a DHD, but still used a dialing computer, likely for security reasons.

The general advantages of a DHD are:

• It will work for thousands of years, even considering stellar drift.
• It has a self-contained power supply.
• It follows a number of safety protocols, ensuring that the journey will be as safe as possible for the passengers, the environment on both ends, and any stellar objects in between.

While in the early months Stargate Command did not have access to a DHD, they do now. It seems the organization prefers their custom dialing computer due to its flexibility and security, but at the cost of safety. Granted, Season 5's "Red Sky" was the last time that having a custom dialing computer actually caused problems (and it prevented problems in Season 7's "Avenger 2.0"), so it seems that they have worked out most of the safety issues.

The short answer?

The DHD was a device made by the people who made the gates, to be used with the gates. Earth's dialing computer was not.

Here is another way of thinking about it. The gate, is like a television. The DHD is the remote control made for it by the manufacturer. Earth's dialing computer... is what a bunch of monkeys would come up with, if they lost the remote to the TV.

The Long answer:

The DHD is a crystal based computer and power supply for the Stargate. It Was created by the Ancients whom had the full technical specifications of the stargate. It uses crystal based technology, and wireless communication to the gate. It has over 200 safety and feedback protocols used to ensure a safe transit through a wormhole, including time outs and redundant power to an incoming matter stream. It communicates to other DHD in order to continually update for new gates and stellar drift.

The SG1 computer though, was a black box reverse engineering of the Stargate. It was done completely by guesswork based on the gates feedback signals. It ignores many of those signals, which causes multiple problems when the SGC forces a connection or when the outbound gate loses power.

Since the Giza gate has no DHD, the SGC had to replicate multiple commands blind, and is presented with problems like interstellar drift, which the DHD was designed to compensate for. It also protects the SGC from virus attacks as it does not communicate with the rest of the Gate network.

The Asgard and Nox have their method of dialing a gate, but due to their status as allies of the Ancients, they were likely provided the technical specifications.