Ping is an ICMP protocol. Basically any system with TCP/IP could respond to ICMP calls if they were not blocked by a firewall or similar filtering mechanism.
Telnet is a remote terminal server application. It must be enabled through system configuration and the server must be up for it to respond.
Ping command sends an ICMP request for reply to a computer and the computers returns the request (ping reply) this is displayed by the ping command along with some statistical information such as how long it did take for a packet to respond. While it is a helpful tool to diagnose problems in intranet it is blocked by most firewalls for security purposes.
Telnet on the other hand a remote access server in that it asks the connecting user authentication data and shell access is granted if the user has a valid account on the target system. UNIX systems and most smart network devices support telnet. But telnet is a very old protocol and it sends usernames and passwords over network in clear text format so it is open to eavesdropping and considered very unsecure.
Nowadays it comes disabled by default for many Unix systems. Most system administrators use SSH (Secure Shell) instead of Telnet. One of the main advantages of SSH over Telnet is all the communication is encrypted and sensitive user data travels encrypted over internet so it is not possible to extract these credentials easily.
Both ping and telnet could be used for diagnosing network problems. Ping simply gets a response and it is what it can do. While it is possible to monitor the output from a given port using telnet.