The single biggest difference between the two modes is that Access Point mode allows you to create a hotspot using the same Wi-Fi card that you are using to access the Internet. In contrast, Ad Hoc mode requires the Internet connection to come from a separate card or device such as an Ethernet card, cellular modem, or a second Wi-Fi card in your computer.
Access Point mode is also faster, allowing speeds up to 54 Mbps in 802.11g, whereas Ad Hoc is generally limited to the 11 Mbps of the older 802.11b standard.
When it is run in Access Point mode, Connectify Hotspot is a real Wi-Fi access point running on your computer. Your laptop or computer is transformed into a Soft AP device. Any device that can connect to a regular access point can connect to Connectify Hotspot, with no special setup or software required.
A number of recent devices do not support Ad Hoc mode (but do support Access Point mode) such as:
Android Devices (there are hacks to get rooted phones on Ad Hoc, but they will not do it out-of-the-box)
Blackberry Devices (running 5.0 or later software)
Amazon Kindles (can see the network but cannot connect)
For the above reasons, we recommend that users use Access Point mode if it is available on their computer.