I sent my manuscript for journal publication last 2 weeks. The status changed from 'editor assigned' to 'peer review' and then right now it is 'under review' What is the difference between 'peer review' and 'under review'?

Probably "peer review" meant they were selecting the anonymous reviewers. Now it is "under review", so they have it and are reading it. Exciting! All the best!

‘Peer review is the process of searching for an appropriate Peer reviewers to go through your paper. ‘Under Review’ is the process where, the peer reviewer is going through your paper. Hence, the status is under review. If you do not have a better network where you can get access to experts to peer review you can always opt for professional help.
Many universities provide a lot of help in this area. Check accordingly.
Here is my recommendation, see if its helpful to you - https://www.enago.com/publication-support-services/peer-review-process.htm

All the best!