Should I drop my paper after finding a recent very similar paper? I have been working on a research paper to apply a technique X to solve certain problem Y. The field is in computer science. Approximately some months ago I started to check it up if there was something similar to my proposal, but I did not find anything at all.

A couple of weeks ago and when I have already finished the tests for my proposal, I decided to check it up again. In my new search, I found an article that was published about two years ago and when I read it, I saw that it was very close to the idea on which I have been working. The differences were very subtle, I was using a simplified version, data was collected in a different manner, and other tiny differences. If we talk in percentages, the differences would be like 20 % between my article and the one I found.

The reasons that I did not find this article in my first search were that it was published in a not so well known journal, and while it has been cited before, it were only self-citations and there was no strong relation between the article and the citing ones.

So what I can do in this situation? I have not based my work on this paper, but I know that I must cite it like a related work. Should I drop my paper or present it and see what the reviewers have to say about it? I would not like to be pointed as a case of plagiarism because of the similarities.
First of all, you are absolutely right that you need to cite the paper. Citations should not be to "previous work you based your work on" but to "the state of the art" - whether or not you explicitly based your work on a previous article. The reader should be able to place your work in the context of what is already known on a topic.

Second, giving any advice will be hard for us, since we don't know the specific situation. If all you add to this previous article is a simpler design and collecting data in a different manner, then your paper likely does not offer a publication-worthy contribution to the state of the art. 

Your best bet is likely to rework your paper heavily. Identify weaknesses or open questions remaining in the previous article and address these. Then you can build on the previous article and expand upon it, and you will have a genuine contribution.

Of course, this will be a lot of work, and much of what you already did may turn out to be wasted effort. That sort of thing unfortunately happens in academia. You will need to analyze your specific situation to find out how much you can salvage.

The positive side is that you have found someone who can meaningfully review your work. And if, as you write, their work has so far mainly been self-cited, they will appreciate some external attention on it, so if you get them to review your paper, you may have a sympathy bonus. Alternatively, you could even contact the authors of the earlier paper and see whether they would be interested in collaborating with you.