In IMP1, heavy infantry gets ranged attacks from the start while regular infantry doesn't until the 3rd generation. That's a good reason for me to use them. Another advantage is that the first 2 generations of regular infantry require iron while heavy infantry requires bronze. Than can also be an advantage if you're short on iron but not on bronze.

You're mixing up IMP 1&2 there o_O

It's possible to take out those Verduns if you are willing to commit the resources and manpower needed. This of course is based on the assumption you are so strong as to be able to raise large numbers of manpower [if you can recruit 20 or 30 or more workers to the city per turn] and that you are not heavily committed elsewhere. It also depends on how much you really want that city. If you have him boxed up, all you need to do is keep an army next to him and he will never stick out his neck. Assuming of course you have already obliterated his navy and merchantmen.

To take a massively fortified city I build lots [and I do mean lots] of scouts and light artillery. You will need a lot because you will lose a lot. I use one or two engineers to knock down the walls. I make a line of scouts and rush them forward. They get wiped out but they draw fire. Next, you rush your light arty forward and hit his heavy arty. Don't simply wear them down, as they will rest up and come back to full power later. Kill as many as possible. Once dead, he cannot rebuild them. You will have to repeat this process several times, but if you have the resources and willingness to use them, you will prevail.

Sometimes the computer will make a tactical error in setting up the fort's defense. He may have lots of heavy arty, but leave them off screen and pile up lots of mobile artillery in the front line. To cope with this, I line up my heavy arty but leave one row of dots empty. This will encourage him to move his light arty up. On my turn, I move my heavy arty forward a row and the next turn I wipe out his light arty. Then I can use my own artillery, light and heavy, to wipe out the rest of his army.

Often, though, I would prefer to wait until a tech progression. Just make sure to leave enough spare arms builds so you can upgrade all or much of your key units.

Any advice for tackling fortified spots in Imperialism 2?

I think Imp 2 improved a lot of areas from the first game (sadly, it was a huge step back in charm and art design), but one thing that got me is they made attacking fortified spots even more impossible than Imp 1.

Forts are cheap and provide free heavy artillery, plus the attacker is now severely limited in how many units can attack at once. I never found a realistic way of taking fortified areas. Even being an entire tech generation ahead was tough.