Tree branch fell on my car, what are some issues which could be caused by this kind of impact?

Hello,

Depending on the damage by the tree, it could just exterior damages without any engine, transmission or suspension issues.

Are you experiencing anything different?

If the impact was severe enough it can cause damage to the suspension. Could have damaged the struts or broken controls arms or other suspension components (if there was significant amount of force)
If you have sunroof it can damage that with an impact from the top.

Also can you upload few pictures? So have a better idea.

Thank you.

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It would depend on the actual damage, but unless there are physical dents on the roof or if it has a sunroof it's unlikely there is any internal damage. Now if you have a sunroof run it a few times just to be sure it works okay, if it does then it should be okay as well. The dent on the hood shouldn't cause any issues unless it damaged the wipers or washer nozzles, again just operate them and if they are okay they should be fine. As to the dent, it would depend on how deep it is and where as most newer vehicles don't have a lot of space under the hood. As long as it isn't hitting or rubbing it shouldn't be an issue to run the car until it is repaired. Have you talked to the insurance company yet? If not I would as that might be covered if the tree was on private property and it may fall under your employers insurance.
Good luck with it.
Repair wise it might be a dent that could easily be pushed out from underneath, but it depends on where it hit. If it worries you that it might be touching an easy way to tell would be to look at where the dent is, then at where that spot is over the engine bay. Now dust that area with some baby powder, you want just a light dusting, then close the hood. Now apply just a bit of pressure to the hood to make it seem like air is pushing it down. Now open the hood and look at the sound deadener under it. If you see a lot of powder transfer you might want to get the dent repaired soon. If you see none then it will be okay until you decide what to do.

Thank you for all of the responses. There's been some pulling of the wheel to the right and that's what I'm worried about with reference to the impact. My commute to work is over 45 minutes and I don't want to run into trouble on the highway. Although there seems to be no damage to the outside of the vehicle other than the bottom plastic part of the driver's side rearview mirror coming off, it's annoying that I was basically condescended to and told nothing's wrong simply because there's no significant physical damage to the outside of my car. Again, I appreciate the responses and the suspension was actually what I worried the most about, so at least reading that gave me some validation for my concerns. I've attached a photo of the 13-ft tree branch that fell from approximately 25 feet above the vehicle and slammed into my car, then rolled onto the ground in front of it. It's not exactly a twig. (Sorry, I tried to rotate the picture numerous times, but the site wouldn't let me turn it right side up.) I'll definitely be getting an estimate elsewhere.

From the picture it does not appear that is sustained too much damage except for some dings and scratches.
For the pulling to the side, have the alignment checked and corrected.
Are you hearing any strange noises or rattles or any other issues?