Assessed value: why so much differences?

Why assessed value very different for the land of one property vs. the same size land of the neighbor property?

If I buy a house, is the assessed value recalculated at that time to a new fair value? Which means that if someone had the house for 30 years, the recorded assessed value is extremely low and it will jump up as soon as I buy a place?

Although I cannot provide an accurate reason for the particular property you have in mind, there can be many reasons for a different assessment for “seemingly” comparable properties. One of the more likely reasons is that the properties have different zoning, especially if one is located on a major road and the other is not. Another reason could be access to the property; if one of the parcels is “landlocked” and the other has road frontage. Another reason might be availability of water/sewer service. Another reason could be one property has deed restrictions that the other does not. Another reason might be that one property is part of some government sponsored “designated protected area” of some kind and the other is not. There are innumerable reasons there could be a difference in assessment, even issues like one owner was “friendly” or related to the assessor or politically connected or even that the assessor was incompetent.