Can use my 3G simcard on my 4G phone? Would that work?

Offcourse You Can use the 4g sim easily. it will enhance the network speed and the other services too like alerts, call quality, MMS, etc etc

Nothing special except the internet speed is faster

Cummon Yaar...

R u kidding with me, Offcourse you can use your 2g sim too.

Hey Offcourse a BIG YES because all the latest 4G LTE Smart Phone will support 3G as well as 2G But WCDMA sim not CDMA, You can use it without a doubt.

What is the advantages of switching a 3G sim card to a 4G phone. Will it be faster? Will I be able to access apps requiring
greater memory?

Yes it will definitely work. My friend is using 3G simcard on 4g phone. It was working smoothly

The 3G SIM will give you access to the 3G & 2G spectrum available in your network, but you will need a higher capacity 128kB 4G nano/micro SIM to access 4G data speeds on your 4GLTE enabled phone.
Once I got my Airtel 4G sim, I continued using the 3G SIM till I got an sms confirmind 4G SIM activation from Airtel when I switched off phone and changed SIMs. Upon switching on phone I had access to blazingly fast 4G LTE speeds.

Even 2g works pranay.

Why not? 3G will work on 4G phones, but 4G won't work on 3G phones.