Exploring options for buying property

I own a house/land outright (as much as a government will let you own anything): I have no mortgage or liens. For examples when comparing options, let's use a nice round number and say that the house is worth $100,000.

I'm interesting in buying a different house/land in another area.

One option would be to use the equity I have in my current house towards the purchase of a second house. I'd put less cash down, end up with 2 houses, have a smaller mortgage than financing the entire price of the new house, but also have a lien on my old house.

Another option would be to sell my current house and then buy the new house. I'd put much less cash down (or none), have a small or non-existant mortgage, and end up with 1 house.

What are financial advantages and disadvantages of those options? Are there other options?

I appreciate your thoughts and insight.