HSRP, or hot standby routing protocol, is very commonly used in the networking world. It is a great solution for providing redundant gateways for a wide array of networks. Over time there have been improvements and changes that have been made as you would expect with any bit of technology. At this point, but HSRP version 1 and version 2 are valid options that can be used. There are similarities between them like the core functionality of what HSRP can do, but there are some differences that you want to be aware of when configuring your network.
Millisecond Timer Values
Millisecond timer values were something that was not advertised in HSRP version 1. In version 2 though, the timer values are advertised to other peer routers to ensure that values are consistent across a standby group. By advertising and learning these values, the routers can learn what the peer devices are using for these timers and then they can be configured to match according to HSRP best practice.
In HSRP version 1, the values that could be used for standby groups ranged from 0 to 255. With networks having so many vlans nowadays for instance, there may be a need to have more than the possible 256 standby group numbers that version 1 could offer you. HSRP version 2 allows the standby groups to be configured with values from 0 to 4095, so definitely some more options there.
HSRP advertisements and communication between devices is sent as multicast traffic. HSRP version 1 uses the multicast address 18.104.22.168. HSRP version 2 uses multicast address 22.214.171.124 for its communication. The reason for this change is around the CGMP, or Cisco Group Management Protocol which is a protocol used to help direct multicast traffic. The problem is that CGMP also uses 126.96.36.199, so issues arose when this was used in a HSRP version 1 network, hence the change to 188.8.131.52 with version 2.
Those three differences are the three most commonly noted differences that I have come across between HSRP version 1 and version 2. By default, version 1 is still the standard in Cisco environments unless you, the one configuring the network, specify that you want to use version 2. Keeping the three reasons above in mind, will help you determine which version will be the best for the specific scenario you are working on.