I'm not a lawyer, but worked in the US for 6 years as a Canadian TN. The green card eligibility is the big differentiator between TN and H1B. Other than that, there isn't much difference that I could tell. One difference between Mexican and Canadian TNs is that Mexican TNs can switch employers without renewing TN, whereas with a Canadian TN you have to reenter the country when you switch employers.
When I was switching jobs I encountered a handful of companies that would consider H1B holders but would not consider TNs because they simply had no experience hiring TNs. I didn't fit their hiring pipeline.
I had some banks give me a rough time when seeking financing for buying a house, who were not familiar with TNs. Wells Fargo were willing to work with me.
Another issue I experienced (I think, my memory is fuzzy on this one) is because various members of my family had passports that expired on different dates (kids get a 3 year passport only), the TN and TD (dependent) were only granted up until the expiry of the passport, so less than 3 years that a TN could be granted for. To avoid costly plane flights I used to drive down to the Mexican border where the border folks were nice enough to renew my TN/TDs.
The US/Mexico border folks don't get many Canadians entering there so they aren't really set up for it. One time they erroneously gave me a Mexican TN authorization which I didn't learn about until I went back there to renew after switching jobs and the border person basically told me to get lost because I had a valid visa. I had to do some quick talking because even though I had a stamp, the visa was invalid because it is up to me to ensure I have valid work authorization.
The website for Canadians entering the US for TN purposes is clearly geared to Canadians entering from the Canadian side of the US, but doesn't explicitly say Canadians shouldn't enter from the Mexican side. It worked out in the end but no guarantees.
My advice: if the company is offering to switch you over to H1B, take them up on it. My employer originally said they would apply to switch me over to an H1B, but when it came time to actually do it, they no longer had budget to do it. This eventually became important because in Texas it affects the ability of your kids to apply for (state?) student loans (I'm fuzzy on this one). I ended up moving back to Canada to be with my kids while they go through university.
The company is asking if I want to apply for a H1B visa this year. Are there any obvious advantages of H1B other than eligibility for green card.
Given the current political climate, is there any projections on how work visas might change going forward.