dhcp is allowing both devices to determine the address to use automatically, instead of you having to program it manually
for surfing the web, email, youtube, it doesn't much matter what address your device has, or how many hops it has to get through. your setup is what's called double NAT - network address translation. The modem is taking the ISP provided address and translating it into a range of addresses (192.168.x.x in the example above). The router takes one of those addresses and then does another translation into a new range (10.10.x.x for example).
this only becomes a problem if you're trying to setup outside access to something, or apps that are very sensitive to timing (twitchy shooters)