Strike is a much stronger word than seems. Strikes is usually more of a revelation/discovery/realisation. It strikes me that my girlfriend might be cheating on me. Seems; Although seems has a similar meaning, it would usually mean something more subtle. It is more associated with making assumptions and suggestions, or what is implied. It seems to me that this year's chemistry curriculum may be harder than the previous year's.

Look up the literal meaning of "to strike (someone)" and you'll get the idea. I think both phrases are suitably formal.

Thank you!

"Seems to me" is much, much more common, at least in the US. I would probably use it 500 times for each time I use "strikes me." There is a subtle difference. "Strikes me" is often used when you just thought of something or you are offering a first impression. "Seems" is used more as a way to introduce an opinion that has been formed over time.