Your question alludes to a paradox:
"azathioprine weakens the immune system ..."
"interferons boost the immune system ...
They seem to have different functions, but they treat the same disease!"
Hence, the latter sentence can easily be paraphrased to: ...they seem to have opposite functions, how can one and the same disease be treated in opposite ways?
This apparent paradox can be resolved to "the immune system" in the statement on azathioprine refering to the adapative immune system and the statement on interferons refering to the innate immune system.
Therefore, those statements should read: "... azathioprine weakens the adaptive immune system, whereas interferon strenghtens the innate immune system." Then, there is no apparent paradox as both statements in their stating opposite mode of actions do not refer to the same.
Still, there is contraction, so the second part of your question, asking about the mechanism of action of the interferon is valid: If interferon does not interfere with the adaptive immune system (i.e. cell division of t- and b-cells) what different, what other, mechanism does it deploy then?
To sum up my internet research: That is not known for certain. However, there are arguments that boosting the interferon part of the immune system dampens the adaptive part of the immune system. Thus, there is a second way to solve the paradox:
Strengthening, "boosting" interferon (interferon medication) dampens t- and b-cell proliferation, in that way "weakens" the adaptive immune system. Both modes of action's ends would be immunosuppression.
"(T)o reduce t- and b-cell count" there would be two ways, then:
Azathioprine interferes with the synthesis of nucleic acids, thus dampening proliferation of t- and b- cells.
Interferon beta might, presumably, do the same, dampening proliferation, by some mechanism that is unknown, but might involve cell signaling (Both mechanisms differ, but might aim at the same).
Against this second part of the answer, and in favour of the validity of your question,
see Wikipedia on Interferon (The following might have given rise to your question):
"IFNs also have various other functions: they activate immune cells, such as natural killer cells and macrophages, and they increase host defenses by up-regulating antigen presentation by virtue of increasing the expression of major histocompatibility complex (MHC) antigens."
Wikipedia's mentioning of killer cells and macrophage (innate immune system), not t- or b-cells (adaptive system) in that context supports the view, see above, that boosting the innate system is able to dampen the adaptive immune system. Think of some virus that, on one hand, is being weak enough to be killed by the innate immune system, but, on the other hand, thus is able to circumvent and evade the adaptive immune system, (i.e. evades immunity, if not autoimmunity, though).
This could imply that any theory assuming that immune suppression could, in the end, be achieved by interferon as well (then, the term "immune suppression" might replace "immune system" in the sentences above) must be restricted to the process of proliferation of t-/b-cells. Proliferation - which is central to azathioprine mechanism - is not mentioned in Wikipedias statement.
Interestingly, Wikipedia on Interferon Beta 1a (not: "Interferon") refers to literature that considers "side-effects" what is considered immunosuppression above:
"Interferon-beta can also reduce numbers of white blood cells (leukopenia), lymphocytes (lymphopenia) and neutrophils (neutropenia), as well as affect liver function.", refering to E.U. Walther, R. Hohlfeld Multiple sclerosis - Side effects of interferon beta therapy and their management. The authors "... discuss the mechanisms and management of the different side effects of IFNβ." Arguably, any results of the medication that are not reduction of known symptoms of the disease might be defined as side effects; thus, this study does not necessarily preclude a theory that - in the sense of your question - assumes that both medications ("dampening and boosting") share the same end: decreasing t- and b-cell count. However, alternatively, the answer to your question might read:
As both medications do not target the same mechanism of the immune system the contradiction is only in words (equivocal use of the term "immune system").
The mechanism of action of interferon beta is not known precisely, cp., e.g.:
Kieseier, The mechanism of action of interferon-β in relapsing multiple sclerosis, CNS Drugs, 2011:
"The mechanism of action of IFNβ is complex, (...) IFNβ appears to directly increase expression and concentration of anti-inflammatory agents while downregulating the expression of proinflammatory cytokines. IFNβ treatment may reduce the trafficking of inflammatory cells across the BBB and increase nerve growth factor production, leading to a potential increase in neuronal survival and repair. IFNβ can also increase the number of CD56bright natural killer cells in the peripheral blood. These cells are efficient producers of anti-inflammatory mediators, and may have the ability to curb neuron inflammation.
The mechanism of action of IFNβ in MS is multifactorial and incompletely understood. (...)"
For azothiprine as the drug of reference cp. Chris' answer to your (related) question:
"... is a purine analog, and the accepted mechanism of action is at the level of DNA. (...) (A)zathioprine (...) can (...) block the pathway of purine synthesis. It is this action that is thought to contribute to its relative specificity to lymphocytes due to their lack of a salvage pathway. However, the effects on the blockade of DNA replication have never fully explained all of the laboratory and clinical findings of azathioprine-induced immunosuppression."
For more research on interferon beta my hint is to do searches for microglia, cp. e.g. Kim et al., Interferon-beta activates multiple signaling cascades in primary human microglia, J Neurochem. 2002
About the possibility of side-effects see e.g.this page
One of the most recent papers relevant to the second part of your question:
Guerrero/Sicotte, Microglia in Multiple Sclerosis: Friend or Foe?
Some excerpts: "For example, new data indicates that there is higher expression of type I interferon and complement genes in gray matter and higher expression of NF-κB inhibitor genes in white matter. (...)
Current disease modifying therapies (DMT) have been shown to have a modest effect on microglia and are divided into indirect and direct effects (two articles refered to). Interferon beta and glatiramer acetate exert an indirect effect by inducing a Th2 shift in lymphocyte profile thereby reducing the pro-inflammatory phenotype of microglia (same reference as above). Interferon beta suppresses interferon gamma induced MHC class II expression on microglia but paradoxically increases the production of inflammatory mediators such as TNF-α, IL-1β, IL-6, and NO." - In my opinion, these recent findings speak in favour of a synergic mode of action of the two drugs you mention: Both seem to reduce t-cells' activity. It is no common knowledge that behind the blood brain barrier there should be none (t cell activity). Non-pathologically, microglial cells, that are equivalent to dendritic cells within the cns - do not even express the MHC essential for inducing (auto-) immunity.