Bagels - Use of Barley Malt Syrup or Non-Diastatic Malt Powder
First, it depends on whether your malt syrup is diastatic or non-diastatic.  (Diastatic contains active enzymes; most syrups are non-diastatic because the process to create syrups usually involves heat that destroys the active enzymes.  However, this is not universally true.)

Diastatic malt's active enzymes help convert starch into sugar.  Some bread or bagel recipes use diastatic malt to speed rising (by converting some starch to sugar quickly to give food to the yeast), as well as to condition the dough and soften the crumb somewhat.  The extra rise can also sometimes lead to a lighter result, and the extra sugar can lead to more flavor and good color.  One has to be careful about adding too much diastatic malt to a dough, because it can weaken it too much and produce a "gummy" texture.  Bagels would likely end up flat and/or with an "undercooked" texture.

Non-diastatic malt is just used for flavor and color.  It has no active enzymes, just a malty flavor which is generally more pronounced than diastatic malt products, due to additional roasting and concentration of flavors not possible at lower temperatures when trying to keep enzymes around in diastatic malt.

A little bit of diastatic malt can be useful in bagel dough.  It will affect texture as well as giving sweetness and color.  Non-diastatic malt (in powder or syrup form) is just about giving sweetness and promoting color.  If you're adding malt to the water to boil bagels, there's no point in using diastatic malt, since the boiling will destroy the enzymes.  If it's all you have on-hand, however, it can still give a malty flavor and promote color when used in larger quantities.

If both of your options are non-diastatic, there's not a huge difference between powder and syrup.  Powder's a bit easier to handle for dough; the syrup is notoriously sticky.  But syrup is sometimes easier to dissolve if you're using it in boiling water.  The more important difference may be in flavor: you specifically mention "barley malt syrup" vs. generic "malt powder."  Some malt powders are made from other grains and may not have the sweetness or complex character of barley malt.  The composition of the malt and its specific flavor (which might be developed in different roasting techniques, even if the same grain is used) can have a greater impact than whether one chooses malt powder vs. syrup.  I do believe, however, that malt syrups often have a darker roast than non-diastatic powders, so that may be useful for more color (especially in the boiling step).  Unless you're buying specialty "dark" malt powder, that is.

Also, the quantity used is most important: in bagels, the contribution of malt to the flavor is usually relatively small (though some would say essential).  Unless you're using a lot of malt, you probably won't notice a significant difference between the types of non-diastatic malt in terms of flavor, texture, etc.