A one-line answer is that a module is 'a minimal stand-alone packaging of a chipset and it's supporting components, on a PCB (Printed Circuit Board) designed to be integrated in a product'
As noted already, a module is typically a small PCB assembly (for example this one). These will contain an ASIC, an oscillator/crystal and usually an antenna. There will also be a handful of discrete components for decoupling etc. Most modules will provide a fairly low speed communication interface, serial, SPI or silimar (but could expose something like a PCIe interface if its relevant).
A chipset consists of one or more ASICs, typically an MCU and an RF section (maybe just the RF section is protocol specific), as well as a software stack.
The most important difference is that a module is likely to be pre-certified (FCC and EU type approval). If you use a module in the way specified by the manufacturer then you can generally avoid having to go through the radio compatibility testing. This approach may preclude you from using a PCB trace antenna, or require that you use a specified trace layout.
If you use a chipset, you need to connect together any relevant parts of the chipset (maybe an analogue and a digital device, maybe an external power amplifier). You need to provide the correct power supply regulation and decoupling, and consider all of the RF layout issues.
A challenge with using a chipset is that the necessary RF test equipment may cost several $100,000 (certainly if you are talking GSM). Without this, you'd need a 3rd party to check your implementation and help you pass the regulatory testing. Even unlicensed bands require that your transmissions remain in the unlicensed band.