Pixel based object recognition, like the name says, works by analyzing the individual pixels of an image. For example: You analyze an image with a lot of different shades of blue and some grey pixels - you might assume that this is the picture of a plane in the sky or a ship in the water.
You could also look for similar pictures by calculating the difference between each individual pixel of 2 images and sum up the differences. The smaller the sum of the difference, the more the pictures look alike.
It depends on your field of work, but those methods will most likely not be good enough for solid image recognition.
You also mentioned neural networks. Of course you can feed the raw pixel data to a neural network, but that's not state of the art. You will almost always use a CNN, which brings me to your next point.
Feature based object recognition, like the name says, tries to extract certain features from an image to perform classification. You use several convolutional layers for analysis before you feed the output to a neural network. This architecture is called CNN (Convolutional Neural Network).
What those features actually look like is hard to explain without showing examples. I recommend to watch the following two lectures from the Stanford course CS231n:
The first one explains image classification in general and shows some implementations for pixel based algorithms. The second one dives into CNNs and explains feature based approaches. I highly recommend to watch the full series, but if you are in a hurry, those 2 lectures will tell you what you want to know.