The difference between traceroute and traceroute remote is that traceroute originates from the LearnMyIP.com webserver and traceroute remote originates from a different server.

Traceroute can be used to show you how a site is physically connected to the Internet. Along the way you will also gain an understanding of how networks inter-connect. Traceroute can be used to determine the specific network route taken to reach a specific remote host.

In a simplified way, this is how traceroute works. Every IP packet can specify how many hops it can go through before it is no longer forwarded on. When a packet is no longer forwarded on, that router just forgets all about it, but it also will usually send out a message to the source host saying, "Hey, sorry, but your packet died here." So, traceroute cleverly manipulates these values so that the first round of packets it sends out to the designated host are specified such that they can only go through one hop before dying. So that first hop gets those packets, sees that it's not supposed to forward them on any further and doesn't, and then sends a message back to the source host telling it that the packets died. When traceroute receives the "your packets died here" message from the router, it knows that's the first hop. It then sends on the second round of packets specifying that they can only go through TWO hops, and the cycle continues. It finishes when it gets a response from the final destination. For each hop, traceroute then displays the RTT, Round Trip Time, or the time difference between when the probe was sent from traceroute and the time the response arrived for each packet.