What is the difference between rheobase and threshold?
Excellent question! The difference is the fact that the rheobase is an example of a threshold measure. The threshold, as you correctly suggest, is the minimal energy (typically current level and not voltage as you suggest) to excite neural tissue. The threshold applies only under the specific experimental parameter settings used. These parameters include the type of electric stimulus (for example, a biphasic current pulse), the electrode configuration (for example, bipolar stimulation) and the stimulus duration (for example 200 microseconds).

The rheobase is the minimal current level needed to excite a tissue when an infinitely long stimulus would be applied. The rheobase is typically found by plotting the threshold level (as defined above) as a function of stimulus duration. The reason for developing this measure is the fact that excitable membranes integrate the injected current, so charge builds up in the tissue. In other words, increasing the pulse duration decreases the threshold current level needed. Hence, the rheobase is a convenient threshold measure because it incorporates the duration parameter of the stimulus. It will still depend on things like electrode configuration and stimulus shape.

The function of the strength-duration relationship can be fitted with a decaying exponential: 

I = Ith / ( 1-e(-t/Tsd) )

where I= Threshold current level; 
Irh = rheobase
t = stimulus duration
Tsd = Tau = exponential

Here is a picture of a theoretical strength-duration curve copied from the wiki page on rheobase. [NB: I find the wiki page needlessly complicated].



The rheobase is the horizontal asymptote. An often used parameter to characterize the function is chronaxie, which is the pulse duration equivalent to two times the rheobase. 

The explanation given here was adapted from the book "Cochlear Implants", chapter 5 "Biophysics and Physiology" by Abbas and Miller, two veterans in cochlear implant electrophysiology. Unfortunately I cannot give you a link to the pdf as I own the work in hard copy.