/24 can be represented many different ways and they all mean the same thing.
As represented by a subnet mask /24 is: 255.255.255.0
As represented by binary /24 is: 11111111.11111111.11111111.00000000. 4 octets of 8 bits each. The 1's represent network bits, there are 24 in total in a /24 network, with 8 host bits left. Only the host bits can change for this particular classful network of /24. Classful network are /8, /16 and /24. Classless network are everything in between from /1 to /32 excluding /8, /16 and /24.
To address your question now.
192.168.1.0 and 192.168.1.1 are indeed in the same network, which I am sure you are aware of.
Typically the first address of a network is the: network address = 192.168.1.0 /24 - based from network range provided in question.
And the last address is typically the: broadcast address = 192.168.1.255 /24
Also note that network addresses must be even numbers. Don't worry though, when you create subnets the first address can be nothing but an even number if you create the subnets correctly.
So the difference is:
192.168.1.0 is the network address
192.168.1.1 is a host address (used to assign to any host IP interface)
Regardless of it's relevance here is a great chart to refer to:http://wb2bwu.org/drupal617/sites/default/files/pictures/VLSM_chart.jpg