In Judaism, are converts accepted as full-fledged Jews or do they have a status in any way different than Jews by birth?
Sincere converts who convert to Judaism according to Jewish Law are full-fledged Jews.
It is nevertheless worth mentioning that Judaism does not require a person to be Jewish in order to gain closeness to G‑d in this world or the next. Torah teachings define beliefs and practices for non-Jews who would thereby be referred to as Righteous Gentiles. This is a major reason why Judaism doesn’t proselytize – a person need not be Jewish to attain righteousness and salvation.
What’s more, since Judaism is such a demanding way of life, and Jews are often subject to hardship and anti-Semitism, gentiles who are interested in converting are actually discouraged from Judaism but rather encouraged to live by the Torah’s Noahide Laws for gentiles. This might be a reason why some people think converts are not fully accepted by Judaism. But this is only before conversion; after proper conversion as above, the convert is a full member of the Jewish People.
That being said, even within Jews by birth there are certain distinctions. For example, there are Cohanim (descendants of Aaron within the Tribe of Levi), Levites (general descendants of the Tribe of Levi), and Israel (descendants of the rest of the Tribes). A convert could not have the status of Cohen or Levi with their specific privileges and responsibilities, but neither could any other Jew become a Cohen or Levi. Rather the convert would have the status of Israel.
The Torah is very emphatic about honoring, sympathizing with, and being sensitive to the convert. G‑d commands many times in the Torah that Jews must love the convert, because the Jewish People knows what it’s like to be a newcomer among a different people. If individual Jews do not accept or show proper empathy for the convert, that’s not coming from Judaism, but rather from their own personal shortcomings.
One possibly sensitive area regarding the position of converts in Judaism is regarding marriage. Even though converts have full Jewish status and are permitted to marry freely with born-Jews, often they marry each other. While in some cases the reason for this might be unfounded prejudice, more often the reasons are social. In this way it may be compared to the many reasons why first generation immigrants often marry people of a similar background. But as with new immigrants, over time converts and their children become fully integrated within the Jewish People at large.