In which circumstance does it make sense establish a site-to-site VPN tunnel instead of investing in routing infrastructure to connect a branch office to headquarters?
There are many factors involved when making this sort of decision. A few of the things to take into account -
How far away is branch office. If the branch office is an outpost geographically it might be easier and cheaper to connect with a VPN.
How important is the work that your company does and how important is the branch office in that work. There are no guarantees on the internet of bandwidth, QOS etc. In times of a major incident the internet can be affected by the number of people accessing it for news etc. Important in this respect can mean
i) financially for the company itself
ii) safety related
iii) security related etc...
What is the level of security your company requires. IPSEC VPN's are secure but some companies are not comfortable with connecting over the internet no matter how secure the technology.
Site-to-site VPN undoubtedly give flexibility when connecting up sites and they are a viable option in many cases but there are no hard and fast rules as to when to use one and when to go for leased lines. It all depends on the needs of the company.