Is the consciousness currently in oneself the same as the consciousness that used to be or in the future will be in oneself? People seem to believe that one's consciousness is a "different" consciousness than that of others, but the "same" consciousness as the one that has been in one's body in the past and the one that will be in one's body in the future. How likely is this to be true? I have almost no knowledge about neuroscience, so please refrain from using much neuroscience jargon if feasible, even if it's at the cost of not giving a thorough explanation. A simple "yes," "no," or "we have no idea" answer would suffice if need be.
Short answer
Consciousness is different between individuals and can change over time.

Background
Nelkin (1997) provides the following definition of consciousness:


  When philosophers and psychologists think about consciousness, they generally focus on one or more of three features: phenomenality (how experiences feel), intentionality (that experiences are "of" something, that experiences mean something), and introspectibility (our awareness of phenomenality and intentionality of experiences). 


Although I am not an expert on the matter I think this definition answers your question. You say People seem to believe that one's consciousness is a "different" consciousness than that of others, but the "same" consciousness as the one that has been in one's body in the past and the one that will be in one's body in the future. Because the phenomenality changes over time, because past experience change the way sensations feel (new versus familiar sensations are very different) your statement that consciousness is constant is invalid. 

The intentionality and most notably the introspectibility makes consciousness a very personal thing, meaning that your statement ...one's consciousness is a "different" consciousness than that of other is valid.  



As a neuroscientist I can tackle the two statements posed on more familiar ground to me. I dare to say that consciousness is not a stable entity, because in the end, sensations, perceptions and awareness of the world are all in your head and generated by neurons. Because psychoneuropharmaca or brain damage can lead to temporary and permanent altered states of consciousness, respectively, the statement that ...the "same" consciousness as the one that has been in one's body in the past and the one that will be in one's body in the future is simply invalid.

From a similar neuroscientific vantage point one's brain is obviously different and physically separate from another person's brain, so we quite frankly are not sharing a consciousness. Consciousness is therefore a personal thing, the more since one can suffer brain damage without another even noticing it, hence one's consciousness is a "different" consciousness than that of others must be true.

Reference
Nelkin N, Philosophy of Science 1993