Difference between push and urgent flags in TCP

They are two vastly different mechanisms.

PSH and the PUSH function

When you send data, your TCP buffers it. So if you send a character it won't send it immediately but wait to see if you've got more. But maybe you want it to go straight on the wire: this is where the PUSH function comes in. If you PUSH data your TCP will immediately create a segment (or a few segments) and push them.
But the story doesn't stop here. When the peer TCP receives the data, it will naturally buffer them it won't disturb the application for each and every byte. Here's where the PSH flag kicks in. If a receiving TCP sees the PSH flag it will immediately push the data to the application.

URG and OOB data

TCP is a stream-oriented protocol. So if you push 64K bytes on one side, you'll eventually get 64k bytes on the other. So imagine you push a lot of data and then have some message that says "Hey, you know all that data I just sent ? Yeah, throw that away". The gist of the matter is that once you push data on a connection you have to wait for the receiver to get all of it before it gets to the new data.
This is where the URG flag kicks in. When you send urgent data, your TCP creates a special segment in which it sets the URG flag and also the urgent pointer field. This causes the receiving TCP to forward the urgent data on a separate channel to the application (for instance on Unix your process gets a SIGURG). This allows the application to process the data out of band.

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As a side note, it's important to be aware that urgent data is rarely used today and not very well implemented. It's far easier to use a separate channel or a different approach altogether.