Do different methods of testing memory for character sequences result in different brain activation? Assume a certain sequence of N characters, displayed on a screen one by one, that should be memorized. The speed of the display is such that it doesn't produce much extra stress. You may assume that the test person might vary display speed within a reasonable range. 

There are several ways to test memory:


A multiple choice format that presents several control sequences.
A control sequence where you should point out differences from a given sequence.
A control sequence, displayed character by character, and the opportunity to push a button as soon as you spot a difference.
Write down what you've memorized.


Do these different ways of testing activate different parts of the brain? If so, where in the brain do these differences occur?
Yes, they do, but not all of these are different methods for testing memory.

Because of the way you've set up the stimulus presentation here, these are effectively all serial verbal working memory tests where the neurocognitive basis may differ or not depending on a number of unstated factors such as stimulus modality or type (e.g., Polyn et al., 2005), that I suspect you are uninterested in. Not all of these tasks would traditionally be considered memory tasks, either--everything involves memory to some extent, and in some cases it is more your choice of a complex stimulus, rather than the test as such, that adds the memory dimension. 

I'm going to focus this answer on the test format (i.e., the retrieval aspect) where possible, and clarify where not. 

1. Serial recognition

A multiple choice format that presents several control sequences along with the stimulus sequence corresponds to a serial matching recognition memory test. Habeck et al. (2005) reported fMRI evidence for increased activity in the cerebellum, thalamus and part of the prefrontal cortex (PFC) regions, plus decreased activity in a broad set of other regions, on a relevant delayed match-to-sample task. 


  Areas of increasing activation across set size were found in the bilateral cerebellum, left lateral PFC (BA 9,46,47), and thalamus. Areas decreasing in activation were more widely distributed and located in parietal, frontal, temporal, and limbic lobes as well as basal ganglia (see detailed listing in Table 3).


2. Parallel recognition

A control sequence where you should point out differences from the stimulus sequence, assuming I am reading that correctly, corresponds to what I'm going to call a parallel recognition memory test, except I don't believe there have been any studies along these lines for serial memory. Usually, this is studied in the context of visual rather than verbal recognition memory, but even here, the neurocognitive evidence is sparse (probably because visual memory gets very little attention compared to verbal memory). Mazumder et al. (2015) reported the following from a "spot-the-difference" task in a recent chapter, which was primarily about novel EEG feature extraction methods:


  The results also suggests that the brain activation is dominant at the frontal, parietal and temporal region.


3. Go/no-go

A control sequence, displayed character by character, and the opportunity to push a button as soon as you spot a difference, corresponds to a complex go/no-go task. This is traditionally an attentional task, which of course involves memory, but does not in itself differentiate between retrieval processes. For posterity's sake, a large meta-analysis of go/no-go fMRI studies reported that neural activity for this task is largely task-dependent, but also finding support for an association with a small set of regions (Simmonds, Pekar and Mostofsky, 2008).


  The simple and complex tasks demonstrated distinct patterns of concurrence, with right dorsolateral prefrontal and inferior parietal circuits recruited under conditions of increased working memory demand.


4. Free recall

Just writing down what you've memorized corresponds to a free recall test (cf. cued recall). Free recall of serial items has often been studied within the context of the primacy and recency effects, referring to the dual tendencies for items at the beginning and end of a sequence to have markedly higher retention and/or faster response times than items in the middle of that sequence. Depending on the item's serial position, different brain regions may show increased activity (Zhang et al., 2003).


  Recalling the middle item was associated with more extensive activation in the left parietal and visual cortex, basal ganglia, and dorsal cerebellum. Recalling items from different serial positions also resulted in different activation time courses in the bilateral primary auditory cortex, left prefrontal cortex and left premotor cortex.


References


Habeck, C., Rakitin, B. C., Moeller, J., Scarmeas, N., Zarahn, E., Brown, T., &amp; Stern, Y. (2005). An event-related fMRI study of the neural networks underlying the encoding, maintenance, and retrieval phase in a delayed-match-to-sample task. Cognitive Brain Research, 23(2), 207-220.
Mazumder, A., Ghosh, P., Khasnobish, A., Bhattacharyya, S., &amp; Tibarewala, D. N. (2015). Selection of Relevant Features from Cognitive EEG Signals Using ReliefF and MRMR Algorithm. In Advancements of Medical Electronics (pp. 125-136). Springer India.
Polyn, S. M., Natu, V. S., Cohen, J. D., &amp; Norman, K. A. (2005). Category-specific cortical activity precedes retrieval during memory search. Science, 310(5756), 1963-1966.
Simmonds, D. J., Pekar, J. J., &amp; Mostofsky, S. H. (2008). Meta-analysis of Go/No-go tasks demonstrating that fMRI activation associated with response inhibition is task-dependent. Neuropsychologia, 46(1), 224-232.
Zhang, D. R., Li, Z. H., Chen, X. C., Wang, Z. X., Zhang, X. C., Meng, X. M., et al. (2003). Functional comparison of primacy, middle and recency retrieval in human auditory short-term memory: An event-related fMRI study. Cognitive Brain Research, 16, 91–98. 10.1016/S0926-6410(02)00223-9