According to what I have read, it seems that erythro isomers are defined in the same way as cis isomers and same holds true for threo and trans isomers. Is there any difference between these two set of terms or they mean one and the same thing?
Imagine an aldol reaction--not condensation--between an enolate of 2,2-dimethylpentan-3-one and benzaldehyde. Two racemic aldols, displayed here as single enantiomers, may be formed,--namely, the anti and syn isomers when the carbon backbones (in red) are drawn in a zig-zag fashion. While these two diastereomers look like trans and cis isomers, respectively, these terms are not applicable because they are a function of how the structures are drawn. Heathcock defined these isomers, respectively, as threo and erythro because the substituents were on the opposite side in the former and on the same side in the latter as in threose and erythrose (see footnote 5 in ref. 1). This choice of terminology was unfortunate in that conversion of the zig-zag conformation to the Fischer projection gives the opposite assignments of erythro and threo. The bottom line is ignore erythro/threo and use anti/syn.
The cis/trans terms are applicable to substituents on rings as is illustrated for the 1,2-dimethylcyclohexanes (red box). Trivial 1,2-disubstituted alkenes are often referred to as cis/trans isomers but Z/E is preferable.