They both have the same number of strings (230), but the strings run horizontally in a grand piano and vertically in an uproight one. In consequance, the grand piano has longer strings and therefore greater resonance and a richer more complex tone.
The above is mostly correct in that 230 is the most common number of strings (the lowest 12 notes have 1 string each, the next 10 have 2 each and the remaining 66 have 3). However, pianos can be built with fewer strings (down to about 215-220, usually extending the 2-string range) or, in rare cases, more (adding a 4th string to the highest notes).
These adjustments are usually based on the desired size and volume, so concert grands tend to have the full 230 (or more) while baby grands or uprights might have slightly fewer.
The string count per note not only provides for a consistent volume and sound quality across the entire range but also influences the effect of the "soft" pedal which slightly shifts the hammers so that fewer strings are struck while playing.
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