I'm having a similar problem, I bought the game and because of my crappy connection it took about 9 - 10 hours to download, I got really excited and launched the game. I noticed in the cinematic it was really laggy, and the gameplay was too. This is the same with GTA: Episodes from Liberty City, my specs are well above the minimum:
OS: Windows Vista - Service Pack 1 / Windows XP - Service Pack 3
Processor: Intel Core 2 Duo 1.8GHz, AMD Athlon X2 64 2.4GHz
Memory: 1.5GB XP / 1.5GB Vista
Graphics: 256MB Nvidia 7900 / 256MB ATI X1900
DirectX Version: DirectX 9.0c Compliant Card
Hard Drive: 16GB of Hard Disc Space
Sound Card: 5.1 Channel Audio Card
Other Requirements: Initial activation requires internet connection; Online play requires log-in to Games For Windows - Live and Rockstar Games Social Club (13+ to register); software installations required including Adobe Flash, DirectX, Microsoft’s .NET Framework 3.0, Games For Windows - LIVE, and Internet Explorer.
OS: Windows 7 Professional
Processor: Intel Core i7 2.8GHz (8 core)
Memory: 4GB DDR3
Graphics: 1GB Nvidia GeForce 9500GT
DirectX Version: DirectX 11
Hard Drive (irrelevant but still): 1TB SATA II 7200RPM
What is going on here?