Isn't wealth redistribution, through whatever means, socialist in nature?

No, wealth redistribution retains class structures and antagonisms but attempts to reduce the inequality that arise from them.

And again, the difference between a society in which social classes are leveled and a classless society is a difference of ends, and I'd imagine that any socialist would believe that a classes made truly level is the same as the elimination of classes in all but name.

I don't think any socialist would think this. Class is not an outcome, qua wealth, but is a power relation, specifically the difference way of interacting with the means of production in a capitalist society. The marxist notion of class is based upon the relation to capital, whereas a liberal notion of class is the amount of wealth or income someone has relative to the rest of the population. For a marxist, wealth inequality is the result of class, but is not itself class. Ownership or lack thereof constitutes class, not wealth. Creating a safety net for workers in a social democratic society would not eliminate the need to sell one's labour power, the extraction and appropriation of value through capital, the constant need for capital accumulation, the power relationship of arising from ownership, and so on.

How are these these different? An end goal is a change that one desires to realize. How is treating the symptoms easily distinguishable from a gradual treatment of the causes through democratic means?

One is capitalism with a human face (with private property, markets, imperatives for growth, etc), and the other is socialist society (socialized production, abolition of the value form, democratic control of the means of production, etc).