Hello,
it might be easiest to understand the difference when you look at it from the perspective of the root switch: the designated port is the port leading away from the root bridge (that is why all forwarding ports on the root are designated ports), while the root port is the forwarding port on the non-root switch pointing towards the root.
Does that make sense ? Here is an extended explanation from KnowledgeNet:
Q:What is the difference between the root port and the designated port on a non-root bridge router. I know how the designated port is determined, but just what does it do?
A:To understand the differences, you must understand the concepts of the root bridge and non-root bridges. Remember that STP creates a logical layer 2 tree of your bridged network. At the bottom of the tree is the root bridge and every other bridge has only one (active) path to the root bridge. To create this tree, STP places ports in one of two states: blocking or forwarding. Determining which port should be blocking and which port should be forwarding can be done automatically by the STP algorithm or you can give it a little help my modifying some attributes such as bridge priority, port cost, port priority, etc. Assuming the root bridge has been selected, each port on the root bridge becomes a designated port and therefore a forwarding port. All other bridges are now considered non-root bridges. To create the tree throughout the topology, STP will go through two different types of elections: root port elections and designated port elections. Every non-root bridge will select its best path to the root bridge. The best path port is called the root port. To stop loops, the third election (called the designated port election) comes into play. Here, each segment in the bridged network elects one designated port and places it in forwarding mode. This designated port is the branch leading away from the root. The idea is, if you only have one port handling traffic on a segment, no loops can be formed.
Have a look at this link as well:
Spanning-Tree - At A Glance
http://www.cisco.com/application/pdf/en/us/guest/netsol/ns24/c643/cdccont_0900aecd800d813a.pdf
Regards,
GP
