Even though I got myself a PSP, I'm beginning to get jealous of people owning a DS. I might start begging my dad to buy my a DS too, but that's quite unlikely. So anyway, here's the comparison:
Graphics
DS: Good graphics
PSP: Good graphics, equal to the DS, if not, only slightly more superior
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Screens
DS: Has two screens. The screens get damaged easily due to the use of the stylus, but that's not very hard to fix once you get those protective layers over the screens...
PSP: Only has one screen. The screen is the largest around compared to any other handheld. It is prone to fingerprints. Even with the protective layer, the fingerprints are visible.
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Model
DS: The DS is smaller than the PSP and is clam-shelled. It fits easily in a pocket. Has 2 should buttons, directional pad, start and select buttons and four other buttons. Has a speaker. Comfortable to hold. The on/off switch slides easily, slide once to off. Close the DS to enter stand-by mode.
PSP: The PSP is larger than the DS. Has two shoulder buttons, directional pad, analog stick, start and select buttons and four other buttons. On/off switch does not slide as smoothly as the DS's. Slide once to enter stand-by mode, slide and hold to switch off. On/off switch can be slid accidentaly while playing games.
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Sound
DS: Has nice sound. I am unsure if speakers or earphones are sold for the DS.
PSP: Again, has nice sound. Contains earphones that come with the package to offer clearer sound when plugged to PSP. Sells speakers seperately (Though I don't think they're any good)
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Capabilities
DS: The DS has a great Wi-Fi system that allows players to look for players worldwide. Internet access and media storage should be coming out in the future.
PSP: The PSP has a Wi-Fi system, though inferior to the DS. Usually only capable of local ad-hoc. Is able to surf the internet and store music, photos and videos. Has game sharing, though NOT available to ALL games.
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Power
DS: Has a long battery life, with a battery lasting to perhaps more than a day.
PSP: Has low battery life. Battery lasts for about 4 hours.
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Games
DS: The DS has 'kiddy' games that not only appeals to kids but the older generation as well. In other words, the games are 'kiddy', but fun. Loads games better then the PSP.
PSP: PSP has games that contains an 'M' rating, suitable to an older audience. Many of the games are imports and remakes of other games from the PS or PS2.
Just to list a few...
DS Games: Mario Kart DS, New Super Mario Bros., Pokemon Diamond/Pearl, Final Fantasy III, Dragon Quest Heroes: Rocket Slime, Metroid Prime: Hunters, Brain Age, The Legend of Zelda: Phantom Hourglass, Contact, Pokemon Mystery Dungeon: Blue Rescue Team, Pokemon Ranger
PSP: Daxter, Syphon Filter: Dark Mirror, Monster Hunter: Freedom, Tekken: Dark Ressurection, Grand Theft Auto: Vice City Stories, Killzone: Liberation, Loco Roco, Lumines, Valkyrie Profile: Lenneth, Fifa Soccer 07, Medal of Honour: Heroes, Metal Gear Solid: Portable Ops