Paper author has not included all suggestions in peer review

The author owns the paper. They should consider any suggestions you make but need not implement any specific recommendation. Give up your "personal" feelings. That isn't your job. Re evaluate the paper on its merits and give it a fair assessment.

You are, of course, allowed to repeat your old suggestions, but it is a mistake to judge it solely on whether all of your suggestions were followed.

Some, but not all, authors will state in a "rebuttal" their reasons for ignoring suggestions. It isn't normally required. Judge the paper as it appears before you. Complain to the editor if you must.

Also consider that some really good suggestions are saved for future work by authors rather than included in the present paper. The work may already be in progress.

It is not your job to "let it through" or not - that is the job of the editor. Write a review that points out that the things you asked for have been changed or not, and say how important you believe these things to be. Certainly for referencing style and spelling/grammar, I would just point this out rather than making a judgement on it. Then let the editor make the decision.

I agree by and large with the two answers already given by Buffy and Ian Sudbery. However I'd like to say something more on who should do what.

It is true that ultimately it is the author who decides what they want to have in the paper, and you cannot enforce them to do anything. It is also true that as a reviewer you do not have the ultimate responsibility to make a decision, this is up to the editor. In particular, although it is nice and helpful to point out formatting, spelling and grammar mistakes, ultimately the job of enforcing correctness in this respect is up to the editor.

It is your job however to decide whether you think that the paper should be published as is, or with further correction, so on the scientific side you have to assess how serious you think the omissions of the author actually are. If you think that the paper should in principle be published because it has something good and original and is by and large correct, however you think that your ignored suggestions are quite important to improve the paper, obviously if the journal has the possibility to run through another cycle, you can state your objections again and say that in your opinion the authors at the very least should reply to them.

As editor I have been in in such situations, I have to make up my mind about this, and it has happened both that I told authors that I think this is really important and they need to address the issues next time, or I occasionally decided that these are side issues in my view or maybe not even justified, and then I wrote to the authors that this is what the reviewer still wants to see and it would be nice if it could be addressed (unless I think it should not), but I wouldn't insist on it.

If of course at this point a final decision of "yes" or "no" is required, you ultimately have to make your mind up about whether you think that this is valuable and should be published in the first place, or whether you think your remaining points are really essential and you recommend to reject if they are not addressed. This may also depend on the level of the journal, you could also think that a paper with these omissions shouldn't be published in a high impact journal, but it could be acceptable elsewhere.